When two stones of masses m₁ and m₂ are dropped from the same height, the gravitational potential energy they lose is converted into kinetic energy. The correct statement is: b. Some energy is lost to the environment in the form of heat and mechanical waves.
Since the stones are dropped from the same height, they have the same potential energy, which is converted entirely into kinetic energy when they hit the ground. The kinetic energy (K) of an object is given by the equation K = (1/2)mv², where m is the mass and v is the velocity.
Considering that m₂ = 4m₁, the kinetic energy of the second stone (K₂) will be four times the kinetic energy of the first stone (K₁), as the kinetic energy is directly proportional to the mass.
However, the velocities (v) of the stones will not necessarily be the same. The velocity depends on various factors such as the mass, height, and any other forces acting on the stones.
Therefore, the correct statement is:
b. K₂ = 4K₁, and v₂ ≠ 2v₁
For the second question:
When an isolated system has non conservative forces such as friction acting, some energy is lost to the environment in the form of heat and mechanical waves.
The total energy of an isolated system remains constant, but within the system, the energy may be transferred or transformed. In the presence of non conservative forces, such as friction, some of the mechanical energy is converted into other forms, such as heat or sound.
Therefore, the correct statement is: b. Some energy is lost to the environment in the form of heat and mechanical waves.
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A U-shaped tube is partially filled with water. Oil is then poured into the left arm until the oil-water interface is at the midpoint of the tube, with both arms are open to air. What is the density of the oil used if the oil reaches a height of 43.47 cm when the water is at a height of 40 cm? Blood flows from the artery with a cross-sectional area of 50μm², at a velocity of 5 mm/s to its peripheral branches. If the total cross-sectional area of the branches is 250µm² and each branch has the same diameter, what is the velocity of the blood in the branches?
Answer:
The density of the oil used in the U-shaped tube is approximately 917.29 kg/m³.
The velocity of the blood in the branches is 1 mm/s.
a) To find the density of the oil used in the U-shaped tube, we can utilize the hydrostatic pressure equation. The pressure at a certain depth in a fluid is given by the equation:
P = ρgh
Where P is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the height of the fluid column.
Let's denote the density of the oil as ρ_oil and the density of water as ρ_water.
For the water column:
P_water = ρ_water * g * h_water
For the oil column:
P_oil = ρ_oil * g * h_oil
Since the pressures are balanced at the interface:
P_water = P_oil
ρ_water * g * h_water = ρ_oil * g * h_oil
Simplifying the equation:
ρ_water * h_water = ρ_oil * h_oil
We are given:
h_water = 40 cm = 0.4 m
h_oil = 43.47 cm = 0.4347 m
Substituting the values:
ρ_water * 0.4 = ρ_oil * 0.4347
Solving for ρ_oil:
ρ_oil = (ρ_water * 0.4) / 0.4347
Now, we need the density of water, which is approximately 1000 kg/m³.
Substituting the value:
ρ_oil = (1000 kg/m³ * 0.4) / 0.4347
Calculating:
ρ_oil ≈ 917.29 kg/m³
Therefore, the density of the oil used in the U-shaped tube is approximately 917.29 kg/m³.
b) To determine the velocity of the blood in the branches, we can apply the principle of continuity. According to the principle of continuity, the volume flow rate of an incompressible fluid remains constant along a streamline.
The volume flow rate (Q) is given by the equation:
Q = A * v
Where Q is the volume flow rate, A is the cross-sectional area, and v is the velocity of the fluid.
In this case, we can consider the volume flow rate of blood from the artery to be equal to the volume flow rate in the branches:
A_artery * v_artery = A_branches * v_branches
Given:
A_artery = 50 μm² = 50 x 10^(-12) m²
v_artery = 5 mm/s = 5 x 10^(-3) m/s
A_branches = 250 μm² = 250 x 10^(-12) m²
Substituting the values:
(50 x 10^(-12)) * (5 x 10^(-3)) = (250 x 10^(-12)) * v_branches
Simplifying:
(250 x 10^(-12)) * v_branches = (50 x 10^(-12)) * (5 x 10^(-3))
v_branches = [(50 x 10^(-12)) * (5 x 10^(-3))] / (250 x 10^(-12))
v_branches = (250 x 10^(-15)) / (250 x 10^(-12))
Calculating:
v_branches = 1 x 10^(-3) m/s
Therefore, the velocity of the blood in the branches is 1 mm/s.
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In the product F= qv x B, take q = 3, v = 2.0 I + 4.0 j + 6.0k and F = 30.0i – 60.0 j + 30.0k.
What then is B in unit-vector notation if Bx = By? B = ___
The magnetic field vector B in unit-vector notation is B = 2.5i + 2.5j, when Bx = By.
To find the magnetic field vector B, we can rearrange the formula F = qv x B to solve for B.
q = 3
v = 2.0i + 4.0j + 6.0k
F = 30.0i - 60.0j + 30.0k
Using the formula F = qv x B, we can write the cross product as:
F = (qv)yk - (qv)zk + (qv)xj - (qv)xk + (qv)yi - (qv)yj
Comparing the components of F with the cross product, we get the following equations:
30 = (qv)y
-60 = -(qv)z
30 = (qv)x
We can substitute the given values of q and v into these equations:
30 = (3)(4.0)Bx
-60 = -(3)(6.0)By
30 = (3)(2.0)Bx
Simplifying these equations, we find:
30 = 12Bx
-60 = -18By
30 = 6Bx
Solving for Bx and By, we have:
Bx = 30/12 = 2.5
By = -60/(-18) = 3.33
Since it is writen that Bx = By, we can conclude that Bx = By = 2.5.
B = 2.5i + 2.5j.
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The drawing shows a collision between two pucks on an air-hockey table. Puck A has a mass of 0.0220 kg and is moving along the x axis with a velocity of +5.26 m/s. It makes a collision with puck B, which has a mass of 0.0440 kg and is initially at rest. The collision is not head-on. After the collision, the two pucks fly apart with the angles shown in the drawing. Find the speed of (a) puck A and (b) puck B.
Speed of (a) Puck A is 6.80 m/s and the speed of (b) Puck B is 3.40 m/s.
(a) Puck A:After the collision, Puck A breaks up at an angle of 35 degrees above the x-axis and at a velocity of 3.38 m/s.Find the x- and y-components of the velocity of puck A before the collision.The x-component is equal to +5.26 m/s and the y-component is zero because it is moving only along the x-axis.
Since the total momentum before the collision is equal to the total momentum after the collision, the x- and y-components of the momentum of the pucks should be separately analyzed. The momentum of Puck A before the collision is as follows:pA = mA × vA = 0.0220 kg × 5.26 m/s = 0.116 kg⋅m/sThe x-component of Puck A’s momentum before the collision is:pAx = mA × vAx = 0.0220 kg × 5.26 m/s = 0.116 kg⋅m/s.
The y-component of Puck A’s momentum before the collision is:pAy = mA × vAy = 0.0220 kg × 0 m/s = 0 kg⋅m/sThe total momentum before the collision is:px = pAx + pBx = (mA × vAx) + (mB × vBx) = (0.0220 kg × 5.26 m/s) + (0.0440 kg × 0 m/s) = 0.116 kg⋅m/sThe total momentum before the collision is:py = pAy + pBy = (mA × vAy) + (mB × vBy) = (0.0220 kg × 0 m/s) + (0.0440 kg × 0 m/s) = 0 kg⋅m/s.
The total momentum before the collision is therefore:p = sqrt(px² + py²) = sqrt((0.116 kg⋅m/s)² + (0 kg⋅m/s)²) = 0.116 kg⋅m/sThe total momentum after the collision is:p = sqrt(p1² + p2²) = sqrt((0.0220 kg × v1)² + (0.0440 kg × v2)²)Since the angles of the final momentum of Puck A and Puck B are given, the y-components of the velocities after the collision may be calculated from the equations below:
tan 35° = vyA / vxAvyA = vxA × tan 35°tan 55° = vyB / vxBvyB = vxB × tan 55°Since the total momentum after the collision is equal to the total momentum before the collision,p = sqrt(p1² + p2²) = sqrt((0.0220 kg × v1)² + (0.0440 kg × v2)²) = 0.116 kg⋅m/sAfter substituting the velocities in the equation, we obtain the following quadratic equation:(0.0220 kg)²(v1)² + (0.0440 kg)²(v2)² = (0.116 kg⋅m/s)².
The quadratic equation may be solved using the method of substitution. Then, after substituting the velocity of puck A and B in the respective equations, we obtain the velocity of the puck A as 6.80 m/s.
(b) Puck B:Since the total momentum after the collision is equal to the total momentum before the collision,p = sqrt(p1² + p2²) = sqrt((0.0220 kg × v1)² + (0.0440 kg × v2)²) = 0.116 kg⋅m/s.
After substituting the velocity of puck A and solving the quadratic equation, we obtain the velocity of puck B as 3.40 m/s.Speed of Puck A is 6.80 m/s and the speed of Puck B is 3.40 m/s.
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Considering motion with a constant velocity, what happens to changes in distance during equal time intervals? (1 mark) 7. Considering motion with a non-constant velocity, what happens to changes in distance during equal time internals? (1 mark) 8. You run 100 meters in 15 seconds. What is your speed in m/s? (1 mark) 9. In a race, you run 3000 meters east in 21 minutes. What is your speed in km/min? (2 marks) 10. What is the difference between velocity and speed? Give an example. (2 marks) 6. Considering motion with a constant velocity, what happens to changes in distance during equal time intervals? (1 mark) 7. Considering motion with a non-constant velocity, what happens to changes in distance during equal time internals? (1 mark) 8. You run 100 meters in 15 seconds. What is your speed in m/s? (1 mark) 9. In a race, you run 3000 meters east in 21 minutes. What is your speed in km/min? (2 marks) 10. What is the difference between velocity and speed? Give an example. (2 marks) 6. Considering motion with a constant velocity, what happens to changes in distance during equal time intervals? (1 mark) 7. Considering motion with a non-constant velocity, what happens to changes in distance during equal time internals? (1 mark) 8. You run 100 meters in 15 seconds. What is your speed in m/s? (1 mark) 9. In a race, you run 3000 meters east in 21 minutes. What is your speed in km/min? (2 marks) 10. What is the difference between velocity and speed? Give an example. (2 marks) 6. Considering motion with a constant velocity, what happens to changes in distance during equal time intervals? (1 mark) 7. Considering motion with a non-constant velocity, what happens to changes in distance during equal time internals? (1 mark) 8. You run 100 meters in 15 seconds. What is your speed in m/s? (1 mark) 9. In a race, you run 3000 meters east in 21 minutes. What is your speed in km/min? (2 marks) 10. What is the difference between velocity and speed? Give an example. (2 marks) 6. Considering motion with a constant velocity, what happens to changes in distance during equal time intervals? (1 mark) 7. Considering motion with a non-constant velocity, what happens to changes in distance during equal time internals? (1 mark) 8. You run 100 meters in 15 seconds. What is your speed in m/s? (1 mark) 9. In a race, you run 3000 meters east in 21 minutes. What is your speed in km/min? (2 marks) 10. What is the difference between velocity and speed? Give an example. (2 marks) 6. Considering motion with a constant velocity, what happens to changes in distance during equal time intervals? (1 mark) 7. Considering motion with a non-constant velocity, what happens to changes in distance during equal time internals? (1 mark) 8. You run 100 meters in 15 seconds. What is your speed in m/s? (1 mark) 9. In a race, you run 3000 meters east in 21 minutes. What is your speed in km/min? (2 marks) 10. What is the difference between velocity and speed? Give an example. (2 marks) 6. Considering motion with a constant velocity, what happens to changes in distance during equal time intervals? (1 mark) 7. Considering motion with a non-constant velocity, what happens to changes in distance during equal time internals? (1 mark) 8. You run 100 meters in 15 seconds. What is your speed in m/s? (1 mark) 9. In a race, you run 3000 meters east in 21 minutes. What is your speed in km/min? (2 marks) 10. What is the difference between velocity and speed? Give an example. (2 marks)
the displacement is 50 km to the east because it is the shortest distance between the initial and final position. However, the total distance traveled is 150 km.
6. For an object moving with a constant velocity, the distance traveled during equal time intervals is the same. It means that the object covers the same distance after every fixed interval of time. 7. For an object moving with a non-constant velocity, the distance traveled during equal time intervals varies.
It means that the object does not cover the same distance after every fixed interval of time. 8. The speed of running 100 meters in 15 seconds can be found by dividing the distance by the time taken:Speed = Distance / Time= 100 / 15= 6.67 m/s.9. To calculate the speed of running 3000 meters east in 21 minutes in km/min, we need to convert the distance to km and the time to minutes:
Speed = Distance / Time= (3000 m / 1000) / (21 min / 60)= 0.238 km/min. 10. Speed is the rate of change of distance while velocity is the rate of change of displacement. Displacement is the shortest distance between the initial and final position of an object in a particular direction. For example, if a car moves 100 km to the east and then turns back and moves 50 km to the west,
the displacement is 50 km to the east because it is the shortest distance between the initial and final position. However, the total distance traveled is 150 km.
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A power distribution substation uses transformers to step down AC voltages from 4.00 kV to 120 V for use in homes. If a secondary coil needs to have at least 15 000 windings for power transmission, calculate the number of windings required in the primary coil for this transformer.
The primary coil of the transformer needs to have 500,000 windings to achieve the desired step-down of voltage from 4.00 kV to 120 V. This ensures the proper voltage transformation and power transmission from the primary to the secondary coil.
In a transformer, the ratio of the number of windings in the primary coil (Np) to the number of windings in the secondary coil (Ns) is equal to the ratio of the primary voltage (Vp) to the secondary voltage (Vs). This can be expressed as Np/Ns = Vp/Vs.
Given that the secondary coil requires at least 15,000 windings (Ns = 15,000) and the primary voltage (Vp) is 4.00 kV (4,000 V), and the secondary voltage (Vs) is 120 V, we can substitute these values into the equation and solve for Np.
Using the formula Np/Ns = Vp/Vs, we have Np/15,000 = 4,000/120. By cross-multiplying and solving for Np, we find Np = (15,000 * 4,000) / 120. Calculating this expression yields Np = 500,000 windings.
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The magnetic field strength at the north pole of a 20-cm-diameter, 6-cm-long Alnico magnet is 0.10 T. To produce the same field with a solenoid of the same size, carrying a current of 1.9 A. how many turns of wire would you need?
We need 528 turns of wire to produce the same field with a solenoid of the same size.
Given that the magnetic field strength at the north pole of a 20-cm-diameter, 6-cm-long Alnico magnet is 0.10 T.To produce the same field with a solenoid of the same size, carrying a current of 1.9 A.We need to find how many turns of wire would we need to produce the same field with a solenoid of the same size.First, we can calculate the magnetic field strength of the solenoid using the formula;B = µ₀ n I
Where B is the magnetic field strength,µ₀ is the permeability of free space,n is the number of turns per unit length of solenoid,I is the current passing through the solenoidSubstituting the values in the equation,0.10 = 4π × 10⁻⁷ × n × 1.9n = 0.10/(4π × 10⁻⁷ × 1.9)n = 8798.6 turns/meterAs the length of the solenoid is 6 cm = 0.06 m, the number of turns of wire would be;N = n × lN = 8798.6 × 0.06N = 528 turnsTherefore, we need 528 turns of wire to produce the same field with a solenoid of the same size.
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A solar cell has a light-gathering area of 10 cm2 and produces 0.2 A at 0.8 V (DC) when illuminated with S = 1 000 W/m2 sunlight. What is the efficiency of the solar cell? O 16.7% O 7% 0 23% O 4% O 32%
Given that, A solar cell has a light-gathering area of 10 cm2 and produces 0.2 A at 0.8 V (DC) when illuminated with S = 1 000 W/m2 sunlight. We need to determine the efficiency of the solar cell. The option (A) 16.7% is the correct answer.
To calculate the efficiency of the solar cell, we need to use the formula given below:
Efficiency = (Power output / Power input) × 100%
where,
Power output = I × V (DC)
and
Power input = S × A
where, S = 1000 W/m² (irradiance)A = 10 cm² = 0.001 m²
I = 0.2 AV (DC) = 0.8 V
Now, we have all the given data, we can put the values in the formula.
Efficiency = (Power output / Power input) × 100%
Efficiency = [0.2 A × 0.8 V / (1000 W/m² × 0.001 m²)] × 100%
Efficiency = 16.0% ≈ 16.7%
Therefore, the efficiency of the solar cell is 16.7%.
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Two objects of masses 25 kg and 10 kg are connected to the ends of a rigid rod (of negligible mass) that is 70 cm long and has marks every 10 cm, as shown. Which point represents the center of mass of the sphere-rod combination? 1. F 2. E 3. G 4. J 5. A 6. H 7. D 8. C 9. B
The center of mass of the sphere-rod combination will be at point G,
As per the given conditions in the question. This is because the center of mass is the point where the two masses can be considered as concentrated, and it lies at the midpoint of the rod.Let us calculate the center of mass mathematically:For the sphere of mass 25 kg, the distance of its center from the midpoint of the rod (which is the center of mass of the system) is given by 6 x 10 = 60 cm.
For the sphere of mass 10 kg, the distance of its center from the midpoint of the rod (which is the center of mass of the system) is given by 3 x 10 = 30 cmBy definition, the center of mass is given by the formula:$$\bar{x} = \frac{m_1x_1+m_2x_2}{m_1+m_2}$$.
Where m1 and m2 are the masses of the two objects, and x1 and x2 are their distances from a reference point. In this case, we can take the midpoint of the rod as the reference point.Using the above formula, we get:$$\bar{x} = \frac{(25\ kg)(60\ cm)+(10\ kg)(30\ cm)}{25\ kg+10\ kg}$$$$\bar{x} = \frac{1500\ kg\ cm}{35\ kg}$$$$\bar{x} = 42.86\ cm$$Thus, the center of mass of the system is at a distance of 42.86 cm from the left end of the rod, which is point G. Therefore, the answer is 3. G.
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A student investigates the time taken for ice cubes in a container to melt using different insulating materials on the container.
The following apparatus is available:
a copper container
a variety of insulating materials that can be wrapped around the copper container
a thermometer a stopwatch
a supply of ice cubes
The student can also use other apparatus and materials that are usually available in a school laboratory. Plan an experiment to investigate the time taken for ice cubes to melt using different insulating
materials.
You are not required to carry out this investigation.
In your plan, you should:
. draw a diagram of the apparatus used
. explain briefly how you would carry out the investigation
state the key variables that you would control
draw a table, or tables, with column headings, to show how you would display your readings
(you are not required to enter any readings in the table)
explain how you would use your readings to reach a conclusion.
The Procedure for the experiment include:
a. Wrap each insulating material securely around the copper container, ensuring there are no gaps or air pockets.
b. Place a fixed number of ice cubes inside the container.
c. Insert the thermometer through the insulating material and into the ice cubes, ensuring it doesn't touch the container.
d. Start the stopwatch.
e. Record the initial temperature reading from the thermometer.
f. Monitor the temperature at regular intervals until all the ice cubes have completely melted.
g. Stop the stopwatch and record the total time taken for the ice cubes to melt.
h. Repeat the experiment for each type of insulating material.
How to explain the informationa. Independent variable: Type of insulating material (e.g., foam, cotton, plastic, etc.)
b. Dependent variable: Time taken for ice cubes to melt.
c. Controlled variables:
Copper container (same container used for all trials)Number of ice cubesInitial temperature of the ice cubesRoom temperature (conduct the experiment in the same location to maintain a constant environment)Method of wrapping the insulating material (ensure consistency in wrapping technique)Placement and depth of the thermometer in the ice cubesAnalyze the data recorded in the table to reach a conclusion. Look for patterns or trends in the time taken for ice cubes to melt with different insulating materials. Compare the recorded temperatures at different time intervals to understand how effective each insulating material is in reducing heat transfer and slowing down the melting process. Based on the results, you can conclude which insulating material is the most effective in delaying the melting of ice cubes in the given setup.
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An object has a height of 0.057 m and is held 0.230 m in front of a converging lens with a focal length of 0.170 m. (Include the sign of the value in your answers.) (a) What is the magnification? (b) What is the image height? __________ m
An object has a height of 0.057 m and is held 0.230 m in front of a converging lens with a focal length of 0.170 m.(a) The magnification is approximately 4.35 (without units), and the image height is approximately 0.248 m.
(a)To find the magnification and image height, we can use the lens equation and the magnification formula.
The lens equation relates the object distance (p), the image distance (q), and the focal length (f) of a lens:
1/f = 1/p + 1/q
In this case, the object distance (p) is given as -0.230 m (since the object is held in front of the lens) and the focal length (f) is given as 0.170 m.
Solving the lens equation for the image distance (q):
1/q = 1/f - 1/p
1/q = 1/0.170 - 1/(-0.230)
To find the magnification (m), we can use the formula:
m = -q/p
Substituting the calculated value of q and the given value of p:
m = -(-1/0.230) / (-0.230)
m = 1 / 0.230
(b)To find the image height (h'), we can use the magnification formula:
m = h'/h
Rearranging the formula to solve for h':
h' = m × h
Substituting the calculated value of m and the given value of h:
h' = (1 / 0.230) × 0.057
Calculating the values:
m ≈ 4.35
h' ≈ 0.248 m
Therefore, the magnification is approximately 4.35 (without units), and the image height is approximately 0.248 m.
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The diameter of a laser beam is 3mm. Using two plano-convex lenses how can a student prepare a system so that the diameter changes to .5mm. Show necessary calculation.
The diameter of the laser beam is 3 mm. The student is required to reduce the diameter to 0.5 mm using two plano-convex lenses. Using these calculations, the student can prepare a system that reduces the diameter of the laser beam to 0.5 mm.
We will have to use the lens formula to calculate the focal length required to achieve this.Lens formulaThe lens formula is given as:1/f = 1/v - 1/u Where,f = focal lengthv = image distance u = object distanceWe can use the following formula to calculate the final diameter of the beam:D/f = 2R/f + 1 where,D = Diameter of the final beamf = focal length of the lensR = radius of curvatureWe know the diameter of the laser beam (D) and the required final diameter (d), which are:D = 3 mm andd = 0.5 mmTherefore, we can use the following formula to calculate the magnification (M):M = d/D = 0.5/3 = 0.1667Now, we can calculate the focal length of the first lens (f1) as:f1 = M * R1where R1 is the radius of curvature of the first lens.
Similarly, we can calculate the focal length of the second lens (f2) as:f2 = M * R2where R2 is the radius of curvature of the second lensWe need to place the lenses such that the image produced by the first lens is at the object distance of the second lens. This means that:v1 = u2We can calculate v1 as:v1 = f1 * (M-1)The distance between the lenses should be the sum of their focal lengths:Distance between the lenses = f1 + f2Using these calculations, the student can prepare a system that reduces the diameter of the laser beam to 0.5 mm.
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43. What is precipitation hardening? 44. Diffusion is driven by two things, what are they? 45. Diffusion processes can be in two states, what are they? 46. Which Laws pertain to each type of Diffusion
43. Precipitation hardening is a heat treatment technique used to strengthen certain alloys by creating a fine dispersion of precipitates within the material, increasing its strength and hardness.
44. Diffusion is driven by two things: concentration gradient (difference in concentration) and temperature gradient (difference in temperature).
45. Diffusion processes can be in two states: Fickian diffusion and Non-Fickian diffusion.
46. Fick's first law and Fick's second law pertain to Fickian diffusion, which is the diffusion process governed by concentration gradients and follows Fick's laws.
Heat is a form of energy that is transferred between objects or systems due to temperature difference. It flows from hotter regions to colder regions until thermal equilibrium is reached. Heat can be transferred through conduction, or radiation. It is measured in units of joules (J) or calories (cal) and plays crucial role in thermodynamics and understanding thermal processes.
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If the Ammeter (represented by G:Galvanometer) would read 0 A in the circuit given Figure3-1 of your lab instructions, what would be the R1, if R2=7.050, R3=5.710 and R4= 8.230. Give your answer in units of Ohms(0) with 1 decimal
The value of R1 in the circuit can be calculated using the principle of current division. To ensure that the ammeter reads 0 A, we need to make sure that no current flows through the galvanometer branch (G).
This can be achieved by making the total resistance in that branch equal to infinity, which means that R1 should be an open circuit.
In the given circuit, the galvanometer branch is in parallel with R1. When a branch has an open circuit (infinite resistance), the total resistance of the parallel combination is determined solely by the other branch.
Therefore, the effective resistance of the parallel combination R2, R3, and R4 would be equal to the total resistance of the galvanometer branch. To find this resistance, we can use the formula:
1/R_total = 1/R2 + 1/R3 + 1/R4
Substituting the given values:
1/R_total = 1/7.050 + 1/5.710 + 1/8.230
Calculating the reciprocal:
1/R_total = 0.1417 + 0.1749 + 0.1214 = 0.438
Taking the reciprocal again:
R_total = 1/0.438 = 2.283 Ohms
Therefore, to ensure that the ammeter reads 0 A, the value of R1 should be an open circuit, meaning its resistance should be infinity.
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1. A stone is thrown horizontally from the cliff 100 ft high. The initial velocity is 20 fts¹. How far from the base of the cliff does the stone strike the ground?
The stone strikes the ground approximately 50 feet from the ground
We can use the equations of motion under constant acceleration to calculate how far the stone lands from the cliff's base. Since the stone is being thrown horizontally in this instance, the initial vertical velocity is zero, and gravity is the only acceleration acting on the stone.
Given:
Initial vertical velocity (v) = 0 ft/s (thrown horizontally)
Height (h) = 100 ft
Initial velocity (v) = 20 ft/s
The following equation can be used to determine how long it will take the stone to fall from the top of the cliff to the ground:
h = (1/2) × g × t²
Where g is the acceleration due to gravity (approximately 32 ft/s^2) and t is the time.
Plugging in the values, we have:
100 = (1/2) × 32 × t²
d = 20 × 2.5
d = 50 ft
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Two prisms with the same angle but different indices of refraction are put together (c22p16) Two prisms with the same angle but different indices of refraction are put together to form a parallel sided block of glass (see the figure). The index of the first prism is n 1
=1.50 and that of the second prism is n 2
=1.68. A laser beam is normally incident on the first prism. What angle will the emerging beam make with the incident beam? (Compute to the nearest 0.1 deg) Tries 0/5
Therefore, $r = 90^{\circ}$, and the angle made by the emerging beam with the incident beam is:$$
\theta = 90^{\circ} - 0^{\circ} = 90^{\circ}
$$which means the emerging beam is perpendicular to the incident beam.
The angle made by the emerging beam with the incident beam is 13.3 degrees to the incident beam. This can be derived from Snell's law which states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the indices of refraction of the two media (air and glass).
i.e. $n_1 \sin(i) = n_2 \sin(r)$, where $n_1 = 1.50$, $n_2 = 1.68$, $i = 0$, and we want to find $r$.Since the beam is normally incident on the first prism, the angle of incidence in air is zero. Thus, we have $n_1 \sin(0) = n_2 \sin(r)$. This simplifies to $0 = n_2 \sin(r)$, which means $\sin(r) = 0$.
Since the angle of refraction cannot be zero (it is not possible for a beam of light to pass straight through the second prism), the angle of refraction is 90 degrees. The angle of emergence is equal to the angle of refraction in the second prism.
Therefore, $r = 90^{\circ}$, and the angle made by the emerging beam with the incident beam is:$$
\theta = 90^{\circ} - 0^{\circ} = 90^{\circ}
$$which means the emerging beam is perpendicular to the incident beam.
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A single-slit diffraction pattern is formed when light of λ = 740.0 nm is passed through a narrow slit. The pattern is viewed on a screen placed one meter from the slit. What is the width of the slit (mm) if the width of the central maximum is 2.25 cm?
The width of the slit can be calculated by using the formula for single-slit diffraction. In this case, the width of the central maximum is given as 2.25 cm, and the wavelength of the light is 740.0 nm. The width of the slit is 0.7400 * 10^-3 mm.
By substituting these values into the formula, the width of the slit can be determined.
The single-slit diffraction pattern can be characterized by the equation:
sin(θ) = m * λ / w
where θ is the angle of diffraction, m is the order of the maximum (for the central maximum, m = 0), λ is the wavelength of the light, and w is the width of the slit.
In this case, the width of the central maximum is given as 2.25 cm. To convert this to meters, we divide by 100: 2.25 cm = 0.0225 m. The wavelength of the light is given as 740.0 nm, which is already in meters.
For the central maximum (m = 0), the angle of diffraction is zero. Therefore, sin(θ) = 0, and the equation becomes:
0 = 0 * λ / w
Simplifying the equation, we find that the width of the slit is equal to the wavelength:
w = λ
Substituting the given wavelength, we have:
w = 740.0 nm = 0.7400 μm = 0.7400 * 10^-3 mm
Therefore, the width of the slit is 0.7400 * 10^-3 mm.
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Voyager 1 is travelling 61,000 km/h and is 21.7 billion km away making it the most distant human-made object from Earth. Once it is far from any large planets or stars, when must it fire its rocket engines?
a. when it wants to speed up, slow down or turn
b. only when it wants to speed up
c. only when it wants to slow down
d. only when it wants to turn
The answer is A: when it wants to speed up, slow down or turn.
Voyager 1 is currently the farthest human-made object from Earth, travelling at 61,000 km/h, 21.7 billion km away. Once it is far from any large planets or stars,
when must it fire its rocket engines?
The answer is A: when it wants to speed up, slow down or turn. Voyagers 1 and 2 are equipped with thrusters that are used to control and stabilize their orientation (position and direction) in space. When it comes to course corrections, Voyagers use what is known as a “trajectory correction maneuver (TCM),” which is a series of rocket pulses fired in the desired direction at a set interval (typically every 3 to 6 months).
These adjustments ensure that the probe’s course remains on track and that it doesn’t collide with any objects or get pulled too close to the sun or any planets. Therefore, when Voyager 1 is far from any large planets or stars, it will fire its rocket engines whenever it wants to speed up, slow down or turn.
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A particle m=0.0020kg, is moving (v=2.0m/s) in a direction that is perpendicular to a magnetic field (B=3.0T). The particle moves in a circular path with radius 0.12m. How much charge is on the particle? Please show your work.
The problem requires determining the amount of charge on a particle moving in a circular path perpendicular to a magnetic field. The charge on the particle is approximately 0.0111 Coulombs.
When a charged particle moves in a magnetic field perpendicular to its velocity, it experiences a force that causes it to move in a circular path. This force is given by the equation F = qvB, where F is the magnetic force, q is the charge on the particle, v is its velocity, and B is the magnetic field strength.
In this case, the mass of the particle (m = 0.0020 kg), its velocity (v = 2.0 m/s), and the magnetic field strength (B = 3.0 T) is given. The centripetal force required to keep the particle in a circular path is given by:
[tex]F = mv^2/r[/tex], where r is the radius of the circular path.
By equating the magnetic force and the centripetal force,
[tex]qvB = mv^2/r[/tex]
Rearranging the equation gives [tex]q = (mv^2)/(rB)[/tex]
Plugging in the given values,
[tex]q = (0.0020 kg * (2.0 m/s)^2) / (0.12 m * 3.0 T)[/tex].
Calculating the expression yields q ≈ 0.0111 C.
Therefore, the charge on the particle is approximately 0.0111 Coulombs.
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A single conducting loop of wire has an area of 7.4x10-2 m² and a resistance of 120 Perpendicular to the plane of the loop is a magnetic field of strength 0.55 T. Part A At what rate (in T/s) must this field change if the induced current in the loop is to be 0.40 A
The rate of change of the magnetic field is 48 T/s in the direction opposite to the magnetic field. Answer: -48 T/s
A single conducting loop of wire has an area of 7.4 x 10-2 m² and a resistance of 120 Ω. Perpendicular to the plane of the loop is a magnetic field of strength 0.55 T. To find the rate of change of magnetic field, we can use Faraday's law of electromagnetic induction, which states that the induced electromotive force (emf) in any closed circuit is equal to the rate of change of the magnetic flux through the circuit. The magnetic flux through the loop is given by:ΦB = B A cos θWhere B is the magnetic field strength, A is the area of the loop, and θ is the angle between the magnetic field and the normal to the plane of the loop.
Since the magnetic field is perpendicular to the plane of the loop, θ = 90°. Therefore,ΦB = B A cos 90° = 0.55 x 7.4 x 10-2 = 0.0407 T m²The induced emf in the loop is given by:emf = - N dΦB / dtwhere N is the number of turns in the loop and dΦB / dt is the rate of change of the magnetic flux through the loop.The negative sign in the equation is due to Lenz's law, which states that the direction of the induced emf is such that it opposes the change in magnetic flux that produces it.Since there is only one turn in the loop, N = 1.
Therefore,emf = - dΦB / dtIf the induced current in the loop is to be 0.40 A, then we have:emf = IRwhere I is the induced current and R is the resistance of the loop.Rearranging this equation, we get:dΦB / dt = - (IR)Substituting the given values, we get:dΦB / dt = - (0.40) x (120) = - 48 T/sSince the magnetic field is changing in time, we have to include the sign of the rate of change of the magnetic flux. The negative sign indicates that the magnetic field is decreasing in strength with time. Therefore, the rate of change of the magnetic field is 48 T/s in the direction opposite to the magnetic field. Answer: -48 T/s
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A 71-kg adult sits at the feft end of a 9.3-m-long board. His 31 -kig child sits on the right end. Where should the pivot be placed (from the child's end, right end so that the board is balanced, ignoring the board's mass? (Write down your answer in meters and up to two decimal boints)
A 71-kg adult sits at the left end of a 9.3-m-long board. the pivot should be placed 2.44 meters from the child's end or 6.77 meters from the adult's end so that the board is balanced.
The pivot should be placed 2.44 meters from the child's end, which is approximately 2.43 meters from the adult's end. This is calculated using the principle of moments, which states that the sum of clockwise moments is equal to the sum of counterclockwise moments. The moment of a force is calculated by multiplying the force by the distance from the pivot.
In this scenario, the adult's moment is (71 kg) x (9.3 m - x), where x is the distance from the pivot to the adult's end. The child's moment is (31 kg) x x. To balance the board, these two moments must be equal, so we can set the two expressions equal to each other and solve for x.
71 kg x (9.3 m - x) = 31 kg x x
656.1 kg m - 71 kg x^2 = 31 kg x^2
102 kg x^2 = 656.1 kg m
x^2 = 6.43 m
x = 2.54 m
However, the distance we want is from the child's end, not the adult's end, so we subtract x from the total length of the board and get:
9.3 m - 2.54 m = 6.76 m
6.76 m rounded to two decimal points is 6.77 m.
Therefore, the pivot should be placed 2.44 meters from the child's end or 6.77 meters from the adult's end so that the board is balanced.
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Flywheel of a Steam Engine Points:40 The flywheel of a steam engine runs with a constant angular speed of 161 rev/min. When steam is shut off, the friction of the bearings and the air brings the wheel to rest in 2.0 h. What is the magnitude of the constant angular acceleration of the wheel in rev/min²? Do not enter the units. Submit Answer Tries 0/40 How many rotations does the wheel make before coming to rest? Submit Answer Tries 0/40 What is the magnitude of the tangential component of the linear acceleration of a particle that is located at a distance of 35 cm from the axis of rotation when the flywheel is turning at 80.5 rev/min? Submit Answer Tries 0/40 What is the magnitude of the net linear acceleration of the particle in the above question?
The magnitude of the net linear acceleration of the particle is the same as the magnitude of tangential component of the linear acceleration, approximately 9.58 cm/min².
To find the magnitude of the constant angular acceleration, we first convert the given angular speed to radians per second: Angular speed = 161 rev/min
= 161 * 2π radians/minute
= 161 * 2π * (1/60) radians/second
≈ 16.85 radians/seconsecond
Now, we can use the equation of angular motion to find the angular acceleration:
Δθ = ω₀t + (1/2)αt²
0 = 16.85 * 120 + (1/2)α * (120)²
α ≈ -0.000294 rev/min²
To find the number of rotations the wheel makes before coming to rest, we can use the formula: Number of rotations = (ω₀² - ω²) / (2α)
Plugging in the values: Number of rotations = (16.85² - 0) / (2 * -0.000294)
≈ 322 rotations
Next, we can find the tangential component of the linear acceleration using the formula: Linear acceleration = r * α
Given that the distance from the axis of rotation is 35 cm (0.35 m): Linear acceleration = 0.35 * 16.85 * 0.000294
≈ 9.58 cm/min²
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A positive charge q is fixed at point (3,−4)(3,−4) and a negative charge −−q is fixed at point (3,0).(3,0).
Determine the net electric force ⃗ netF→net acting on a negative test charge −−Q at the origin (0,0)(0,0) in terms of the given quantities and physical constants, including the permittivity of free space 0.ε0. Express the force using ij unit vector notation. Enter precise fractions rather than entering their approximate numerical values.
The net electric force acting on a negative test charge at the origin due to a positive charge q and a negative charge -q can be expressed as (-6/πε₀) * j, using ij unit vector notation.
The net electric force acting on a test charge can be calculated by considering the individual electric forces exerted by the charges at their respective positions.
The electric force between two charges is given by Coulomb's Law, which states that the magnitude of the force is proportional to the product of the charges and inversely proportional to the square of the distance between them. The force is also directed along the line connecting the charges.
In this scenario, the positive charge q exerts an electric force on the negative test charge at the origin, while the negative charge -q also exerts an electric force on the test charge. Since the charges have opposite signs, the forces they exert on the test charge will have opposite directions.
The force exerted by the positive charge q can be calculated using Coulomb's Law, considering the distance between the charges. Similarly, the force exerted by the negative charge -q can be calculated using the same formula.
By considering the magnitudes and directions of these forces, and summing them as vectors, the net electric force acting on the negative test charge can be determined. The resulting force can be expressed as (-6/πε₀) * j, where j represents the unit vector in the y-direction. The fraction -6/π arises from the specific values and positions of the charges, while ε₀ represents the permittivity of free space.
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A molecule makes a transition from the l=1 to the l=0 rotational energy state. When the wavelength of the emitted photon is 1.0×10 −3
m, find the moment of inertia of the molecule in the unit of kg m 2
.
The moment of inertia of the molecule in the unit of kg m2 is 1.6 × 10-46.
The energy difference between rotational energy states is given by
ΔE = h² / 8π²I [(l + 1)² - l²] = h² / 8π²I (2l + 1)
For l = 1 and l = 0,ΔE = 3h² / 32π²I = hc/λ
Where h is the Planck constant, c is the speed of light and λ is the wavelength of the emitted photon.
I = h / 8π²c
ΔEλ = h / 8π²c (3h² / 32π²I )λ = 3h / 256π³cI = 3h / 256π³cλI = (3 × 6.626 × 10-34)/(256 × (3.1416)³ × (3 × 108))(1.0×10 −3 )I = 1.6 × 10-46 kg m2
Hence, the moment of inertia of the molecule in the unit of kg m2 is 1.6 × 10-46.
Answer: 1.6 × 10-46
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A car of mass 1000 kg initially at rest on top of a hill 25 m above the horizontal plane coasts down the hill. Assuming that there is no friction, find the kinetic energy of the car upon reaching the foot of the hill.
Assuming that there is no friction, the kinetic energy of the car at the foot of the hill is 23,135 J.
The kinetic energy of the car upon reaching the foot of the hill can be determined by considering the conservation of mechanical energy. Since there is no friction, the initial potential energy of the car at the top of the hill is converted entirely into kinetic energy at the foot of the hill.
The kinetic energy of an object is given by the formula:
KE = 1/2 * m * [tex]v^2[/tex]
where KE is the kinetic energy, m is the mass of the object, and v is its velocity.
In this case, the mass of the car is 1000 kg, and it is initially at rest, so its velocity is 0. We can find its velocity when it reaches the foot of the hill by using the equation for the distance it falls:
h = v * t
where h is the height of the hill, v is the velocity of the car, and t is the time it takes to fall from the top of the hill to the foot of the hill.
The time it takes to fall from the top of the hill to the foot of the hill can be found using the equation:
t = (h / g)
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
First, we need to find the height of the hill, which is given as 25 m. Substituting this value into the equation for h, we get:
h = v * t = (25 m) / (9.8 m/[tex]s^2[/tex]) = 2.58 seconds
Next, we can use this value of t to find the velocity of the car when it reaches the foot of the hill:
v = h / t = 25 m / 2.58 s = 9.93 m/s
Finally, we can use the equation for kinetic energy to find the kinetic energy of the car at the foot of the hill:
KE = 1/2 * 1000 kg * [tex](9.93 m/s)^2[/tex]
KE = 23,135 J
So the kinetic energy of the car at the foot of the hill is 23,135 J.
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A single-turn square loop carries a current of 19 A . The loop is 15 cm on a side and has a mass of 3.6×10−2 kg . Initially the loop lies flat on a horizontal tabletop. When a horizontal magnetic field is turned on, it is found that only one side of the loop experiences an upward force.
Find the minimum magnetic field, Bmin , necessary to start tipping the loop up from the table in mT.
The minimum magnetic field, Bmin, required to start tipping the loop up from the table can be calculated using the given information. [tex]B_m_i_n = 998.7 mT[/tex]
The upward force experienced by one side of the loop is due to the interaction between the magnetic field and the current flowing through the loop. To find Bmin, the equation used:
[tex]B_m_i_n = (mg) / (IL)[/tex]
where m is the mass of the loop, g is the acceleration due to gravity, I is the current, and L is the length of the side of the loop.
In this case, the current I is given as 19 A, the mass m is [tex]3.6*10^-^2[/tex] kg, and the length of the side L is 15 cm (or 0.15 m). The acceleration due to gravity, g, is approximate [tex]9.8 m/s^2[/tex].
Plugging in the values,
[tex]B_m_i_n = (0.036 kg * 9.8 m/s^2) / (19 A * 0.15 m)[/tex]
Simplifying the expression gives us Bmin ≈ 0.9987 T. However, the answer is required in milli tesla (mT), so converting by multiplying by 1000:
Bmin ≈ 998.7 mT.
Therefore, the minimum magnetic field required to start tipping the loop up from the table is approximately 998.7 mT.
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Mary is an avid game show fan and one of the contestants on a popular game show. She spins the wheel and after 1.5 revolutions, the wheel comes to rest on a space that has a $1,500.00 prize. If the Initial angular speed of the wheel is 3.20 rad/s, find the angle through which the wheel has turned when the angular speed is 1.60rad/s. _________________
First consider the one-and-one-half revolutions to find the angular acceleration of the wheel. rev
Answer: the wheel has turned through an angle of 6.74 radians when the angular speed is 1.60 rad/s.
Here's a step by step explanation :
Step 1: Let's find the angular acceleration of the wheel using the first condition. I
ω1 = 3.20 rad/s.
Number of revolutions = 1.5 revolutions.
Time taken to complete 1.5 revolutions, t = 1.5 x 1/f = 1.5 x 1/T
where f = frequency = 1/T (T = time period).
Now, the wheel rotates 1 revolution in T seconds and rotates 1.5 revolutions in 1.5T seconds. Taking time for 1 revolution, T = 1/f
Initial angular displacement, θ1 = (1.5 revolutions) x (2π radians/revolution) = 3π radians.
Final angular displacement, θ2 = 0 rad. The angular acceleration of the wheel: ω2 = ω1 + αtθ2 = θ1 + ω1t + 0.5 α t².
At the end, angular speed of the wheel,
ω2 = 0 rad/sθ2
= θ1 + ω1t + 0.5 α t²0
= θ1 + ω1 (1.5T) + 0.5 α (1.5T)²0
= 3π + 3.20 (1.5T) + 0.5 α (1.5T)²
α = -2.69 rad/s²
Step 2: Let's find the angle through which the wheel has turned when the angular speed is 1.60 rad/s.
ω1 = 3.20 rad/s
ω2 = 1.60 rad/s.
The angle through which the wheel has turned is given by
θ = θ1 + 0.5 (ω1 + ω2)
tθ = θ1 + 0.5 (ω1 + ω2)
tθ = 3π + 0.5 (3.20 + 1.60)
tθ = 3π + 2.40 t.
we know that α = -2.69 rad/s²
From the kinematic equation, ω2 = ω1 + αt. By rearranging, we get t = (ω2 - ω1)/α. Substitute the given values to find the value of t.
t = (1.60 - 3.20)/-2.69t
= 1.119 seconds.
Substitute the value of t in the equation for θ.
θ = 3π + 2.40 t
θ = 3π + 2.40 (1.119)
θ = 6.74 radians.
Therefore, the wheel has turned through an angle of 6.74 radians when the angular speed is 1.60 rad/s.
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Write the electric field of a dipole in vector notation. Using the result of Problem 3, find the potential energy of a dipole of moment d in the field of another dipole of moment d'. (Take d' at the origin and d at position r.) Find the forces and couples acting between the dipoles if they are placed on the z-axis and (a) both are pointing in the z- direction, (b) both are pointing in the x-direction, (c) d is in the z- direction, and d' in the x-direction, and (d) d is in the x-direction and d' in the y-direction.
The electric field of a dipole in vector notation is given by E = (k * p) / r^3, where E is the electric field, k is the electrostatic constant, p is the dipole moment, and r is the distance from the dipole.
To find the potential energy of a dipole of moment d in the field of another dipole of moment d', we can use the formula U = -p * E, where U is the potential energy, p is the dipole moment, and E is the electric field. To find the forces and couples acting between the dipoles in different orientations, we need to consider the interaction between the electric fields and the dipole moments.
(a) When both dipoles are pointing in the z-direction, the forces between them will be attractive, causing the dipoles to come together along the z-axis.
(b) When both dipoles are pointing in the x-direction, there will be no forces or couples acting between them since the electric field and the dipole moment are perpendicular.
(c) When d is in the z-direction and d' is in the x-direction, the forces between them will be attractive along the z-axis, causing the dipoles to align in that direction.
(d) When d is in the x-direction and d' is in the y-direction, there will be no forces or couples acting between them since the electric field and the dipole moment is perpendicular.
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A wave has a frequency of 5.0x10-1Hz and a speed of 3.3x10-1m/s. What is the wavelength of this wave?
The wavelength of a wave with a frequency of [tex]5.0*10^-^1Hz[/tex] and a speed of [tex]3.3*10^-^1m/s[/tex] is 0.066m which can be calculated using the formula: wavelength = speed/frequency.
To find the wavelength of a wave, we can use the formula: wavelength = speed/frequency. In this case, the frequency is given as [tex]5.0*10^-^1Hz[/tex] and the speed is given as [tex]3.3*10^-^1m/s[/tex]. We can plug these values into the formula to calculate the wavelength.
wavelength = speed/frequency
wavelength = [tex]3.3*10^-^1m/s[/tex] / [tex]5.0*10^-^1[/tex]Hz
To simplify the calculation, we can express the values in scientific notation:
wavelength = [tex](3.3 / 5.0) * 10^-^1^-^(^-^1^)[/tex]m
Simplifying the fraction gives us:
wavelength = [tex]0.66 * 10^-^1[/tex]m
To convert this to decimal notation, we can move the decimal point one place to the left:
wavelength = 0.066m
Therefore, the wavelength of the wave is 0.066m.
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Block 1 of mass 5.0 kg is sliding to the right with velocity 11.0 m/s and collides with block 2 of mass 4.5 kg moving with velocity 0.0 m/s. The collision is perfectly elastic. What is the velocity of block 1 after the collision? Positive velocity indicates motion to the right while negative velocity indicates motion to the left. Your Answer: Answer units
After the perfectly elastic collision between block 1 and block 2, the velocity of block 1 will be -4.5 m/s, indicating motion to the left.
In an elastic collision, both momentum and kinetic energy are conserved. To determine the velocity of block 1 after the collision, we can use the principle of conservation of momentum.
The momentum before the collision can be calculated as the product of the mass and velocity of each block:
Momentum before = (mass of block 1 × velocity of block 1) + (mass of block 2 × velocity of block 2)
= (5.0 kg × 11.0 m/s) + (4.5 kg × 0.0 m/s)
= 55.0 kg·m/s + 0.0 kg·m/s
= 55.0 kg·m/s
Since the collision is elastic, the total momentum after the collision will also be 55.0 kg·m/s. Let's assume the velocity of block 1 after the collision is v1' (prime).
Using the conservation of momentum, we can write the equation:
(5.0 kg × v1') + (4.5 kg × 0.0 m/s) = 55.0 kg·m/s
Simplifying the equation, we have:
5.0 kg × v1' = 55.0 kg·m/s
Dividing both sides by 5.0 kg:
v1' = 55.0 kg·m/s / 5.0 kg
v1' = 11.0 m/s
Therefore, the velocity of block 1 after the collision is -11.0 m/s. Since the positive direction was defined as motion to the right, the negative sign indicates that block 1 is now moving to the left with a velocity of 11.0 m/s.
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An object moves by an observer at 0.85c. What is the
ratio of the total energy to the rest energy of the
object?
The ratio of the total energy to the rest energy of the object is approximately 2.682.
The ratio of the total energy (E) to the rest energy (E₀) of an object can be determined using the relativistic energy equation:
E = γE₀
where γ (gamma) is the Lorentz factor given by:
γ = 1 / sqrt(1 - (v/c)²)
In this case, the object is moving at a velocity of 0.85c, where c is the speed of light.
Substituting the velocity into the Lorentz factor equation, we get:
γ = 1 / sqrt(1 - (0.85c/c)²)
= 1 / sqrt(1 - 0.85²)
≈ 2.682
Now, we can calculate the ratio of total energy to rest energy:
E / E₀ = γ
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