1True or false
The crew of a fast-moving spaceship notice nothing unusual about the length of their own ship.
Show answer
Answer: True
In its own rest frame the ship has its proper length; only observers moving relative to it measure it shorter.
!Common mistakeSome think the crew would feel squashed; nothing changes in your own rest frame.
2True or false · ★ Challenge
Doubling the kinetic energy of an electron that is already moving at 0.99c roughly doubles its speed.
Show answer
Answer: False
Its speed can at most rise from 0.99c towards c; the extra energy mostly raises γ (and the momentum), hardly the speed.
!Common mistakeUsing KE = ½mv² (v ∝ √KE) at this speed gives the wrong idea; near c, speed barely changes with energy.
3True or false
The relativistic mass γm₀ of a fast-moving electron, as measured in the laboratory, is larger than its rest mass.
Show answer
Answer: True
γ > 1 for any speed, so γm₀ > m₀; in particle accelerators fast particles behave as if heavier.
!Common mistakeSome think the electron gets bigger or gains particles; it is its inertia (resistance to acceleration) that increases.
4True or false · ★ Challenge
According to the first postulate, an experiment done inside a train moving at constant velocity gives exactly the same result as on the platform, so no experiment inside can detect the train's uniform motion.
Show answer
Answer: True
The laws of physics are the same in every inertial frame, so there is no special "at rest" frame to compare with.
!Common mistakeSome think a very sensitive instrument could detect uniform motion; the postulate says no experiment can, however sensitive.
5Multiple choice · ★ Challenge
In the Michelson–Morley experiment, what result was expected if the ether existed, and what was actually observed?
- ANo fringe shift; a large shift was observed
- BLight slower in one arm, which was then confirmed
- CThe fringes vanishing at noon; they did vanish
- DA fringe shift when the apparatus was turned; no shift was seen
Show answer
Answer: D. A fringe shift when the apparatus was turned; no shift was seen
Motion through the ether should change the travel times along the two arms, so rotating the apparatus should shift the fringes. No shift (a "null result") was found.
!Common mistakeChoosing the reversed option mixes up the prediction and the result; the famous outcome was the null result.
6True or false
The Michelson–Morley experiment used the interference of two light beams that travelled along arms at right angles to each other.
Show answer
Answer: True
The interferometer splits one beam into two perpendicular paths and recombines them; any change in travel time shows as a fringe shift.
!Common mistakeSome think it timed light directly with clocks; it compared two paths using interference, which is far more sensitive.
7Multiple choice · ★ Challenge
Lightning strikes both ends of a moving train at the same time according to an observer standing on the platform beside the middle of the train. What does a passenger sitting at the middle of the train see?
- AThe front strike first, as she moves towards its light
- BBoth at once, because the strikes were simultaneous
- CThe rear strike first, as its light catches up with her
- DNeither one, as light cannot catch a moving train
Show answer
Answer: A. The front strike first, as she moves towards its light
In the platform frame she moves towards the light from the front, so it reaches her first; since light has speed c in her frame too, she concludes the front strike happened first.
!Common mistakeChoosing 'both at once' assumes simultaneity is absolute; events simultaneous in one frame need not be in another.
8True or false
In a non-inertial frame, objects seem to accelerate with no real force acting on them, for example a bottle sliding forward on the floor of a braking bus.
Show answer
Answer: True
Seen from the ground the bottle simply keeps moving (Newton's first law) while the bus slows; in the bus frame it appears to be pushed by a "fictitious" force.
!Common mistakeSome say a real forward force acts on the bottle; nothing pushes it, the bus decelerates under it.
9Fill in the blank · ★ Challenge
A boat moves upstream at 8.0 m/s relative to the water, in a river flowing at 3.0 m/s. Its speed relative to the bank is ______ m/s.
Show answer
Answer: 5.0
Upstream the current opposes the boat: 8.0 − 3.0 = 5.0 m/s.
!Common mistakeAdding to get 11 m/s ignores that the boat is moving against the current.
10True or false
The Lorentz factor γ can never be less than 1.
Show answer
Answer: True
1 − v²/c² is at most 1, so its square root is at most 1 and γ = 1/√(1 − v²/c²) is at least 1.
!Common mistakeUsing γ = √(1 − v²/c²) by mistake gives values below 1 and makes moving clocks run fast.
11Multiple choice · ★ Challenge
A proton is accelerated until its kinetic energy is three times its rest energy. What is its speed?
- A0.97c
- B0.94c
- C2.4c
- D0.75c
Show answer
Answer: A. 0.97c
(γ − 1)m₀c² = 3m₀c², so γ = 4; v = c√(1 − 1/γ²) = c√(15/16) ≈ 0.97c.
!Common mistakeChoosing 2.4c uses ½m₀v² = 3m₀c², which ignores relativity and breaks the speed limit; 0.94c takes γ = 3, forgetting to add the rest energy.
12Fill in the blank
Two events that happen at the same time AND at the same place are simultaneous for ______ observers.
Show answer
Answer: all
Disagreement about simultaneity only arises for events separated in space.
!Common mistakeAnswering "only stationary" overstates relativity: events at the same point are simultaneous for everyone.
13Multiple choice · ★ Challenge
Muons in a beam are measured in the laboratory to live 15.6 μs on average, while their proper lifetime is 2.2 μs. What is their Lorentz factor?
- A7.1
- B0.14
- C34
- D13.4
Show answer
Answer: A. 7.1
Δt = γΔt₀, so γ = 15.6 ÷ 2.2 ≈ 7.1 (they move at about 0.99c).
!Common mistakeChoosing 0.14 divides the wrong way; the dilated lab time is the longer one, so γ = longer ÷ shorter > 1.
14Fill in the blank
The total energy γm₀c² of a moving particle equals its rest energy plus its ______ energy.
Show answer
Answer: kinetic
γm₀c² = m₀c² + KE.
!Common mistakeAnswering "potential" confuses the energy of motion with stored energy.
15Fill in the blank · ★ Challenge
In an experiment, a proton beam moving at 0.70c meets head-on an electron beam moving at 0.50c (both measured in the laboratory). Using the relativistic addition formula, the speed of the electrons relative to the protons is ______c.
Show answer
Answer: 0.89
u = (0.70 + 0.50) ÷ (1 + 0.70 × 0.50) = 1.20 ÷ 1.35 ≈ 0.89c.
!Common mistakeAdding to get 1.2c breaks the speed limit; the relative speed of two objects is always less than c.
16Multiple choice · ★ Challenge
A cylindrical fuel tank 10 m long and 2.0 m in diameter moves along its axis at 0.80c. What dimensions does an observer at rest measure?
- A10 m long and 1.2 m wide
- B6.0 m long and 2.0 m wide
- C6.0 m long and 1.2 m wide
- D17 m long and 2.0 m wide
Show answer
Answer: B. 6.0 m long and 2.0 m wide
Only the length along the motion contracts: 10 ÷ 1.67 = 6.0 m; the diameter (perpendicular) stays 2.0 m.
!Common mistakeChoosing "6.0 m long and 1.2 m wide" contracts the diameter too; perpendicular lengths are unchanged.
17Fill in the blank
The instrument used by Michelson and Morley, which splits a light beam into two perpendicular paths and then recombines them, is called an ______.
Show answer
Answer: interferometer
An interferometer detects tiny differences in path length by the interference fringes they produce.
!Common mistakeAnswering "spectrometer" names an instrument that measures wavelengths, not path differences.
18Multiple choice
Which of these observers is in a NON-inertial frame of reference?
- AA passenger in a train at a steady 20 m/s on a straight track
- BA referee standing still on a football pitch
- CA passenger in a bus that is braking sharply
- DAn astronaut drifting in deep space with the engines off
Show answer
Answer: C. A passenger in a bus that is braking sharply
An inertial frame is at rest or moves at constant velocity; a braking bus is decelerating, so its frame is non-inertial.
!Common mistakeChoosing the train confuses "moving" with "accelerating"; steady straight-line motion is still an inertial frame.
19Fill in the blank · ★ Challenge
A spacecraft travels to a star 4.0 light-years from Earth at 0.80c. According to Earth clocks the one-way trip takes ______ years.
Show answer
Answer: 5.0
t = distance/speed = 4.0 light-years ÷ 0.80c = 5.0 years (the crew age only 5.0 × 0.60 = 3.0 years).
!Common mistakeDividing by γ here mixes up the frames; in the Earth frame you simply use distance ÷ speed.
20Fill in the blank · ★ Challenge
The mass defect of a helium-4 nucleus is 5.0 × 10⁻²⁹ kg. Its binding energy is ______ × 10⁻¹² J. (c = 3.0 × 10⁸ m/s)
Show answer
Answer: 4.5
E = Δm c² = 5.0 × 10⁻²⁹ × 9.0 × 10¹⁶ = 4.5 × 10⁻¹² J.
!Common mistakeMultiplying by c (not c²) gives 1.5 × 10⁻²⁰ J, far too small.
21Multiple choice
What was the 'ether' that most nineteenth-century physicists believed in?
- AThe gas that fills the upper layers of the atmosphere
- BA medium filling space through which light was thought to travel
- CA type of radiation given out by very hot bodies
- DThe vacuum inside a cathode-ray tube
Show answer
Answer: B. A medium filling space through which light was thought to travel
Since sound and water waves need a medium, it was assumed light also needed one, filling all of space.
!Common mistakeChoosing 'the upper atmosphere' is wrong: the ether was supposed to fill even empty space between the planets and stars.
22Short answer · ★ Challenge
Explain why the null result of the Michelson–Morley experiment supports Einstein's second postulate.
Show answer
Model answer: If light moved through an ether, its speed relative to the Earth would depend on the direction of the Earth's motion, and the fringes would shift when the apparatus was turned. No shift was found in any direction or season, so light has the same speed in all directions for the moving Earth – exactly what the second postulate states (and there is no need for an ether).
!Common mistakeSaying the experiment "proved light is a wave" is wrong; it was about whether the speed of light depends on the motion of the observer.
23Multiple choice
Which expression gives the kinetic energy of a particle moving at any speed?
- AKE = γm₀c²
- BKE = ½m₀v² at all speeds
- CKE = (γ − 1)m₀c²
- DKE = m₀c²/γ
Show answer
Answer: C. KE = (γ − 1)m₀c²
Total energy γm₀c² minus rest energy m₀c² leaves the kinetic energy (γ − 1)m₀c².
!Common mistakeChoosing γm₀c² gives the TOTAL energy, which includes the rest energy as well as the kinetic energy.
24Multiple choice · ★ Challenge
A ship moving at 0.50c away from Earth fires a probe forwards at 0.50c relative to the ship. Using u = (u′ + v)/(1 + u′v/c²), what is the probe's speed relative to Earth?
- A0.80c
- B1.0c
- C0.50c
- D0.67c
Show answer
Answer: A. 0.80c
u = (0.50 + 0.50) ÷ (1 + 0.25) c = 1.0 ÷ 1.25 c = 0.80c.
!Common mistakeChoosing 1.0c is the Galilean sum; the relativistic denominator always keeps the result below c.
25Multiple choice
Which formula correctly gives the relativistic addition of a velocity u′ (measured in a frame) to the velocity v of that frame?
- Au = (u′ + v)/(1 − u′v/c²)
- Bu = (u′ + v)/(1 + u′v/c²)
- Cu = (u′ − v)/(1 + u′v/c²)
- Du = (u′ + v)(1 + u′v/c²)
Show answer
Answer: B. u = (u′ + v)/(1 + u′v/c²)
The sum u′ + v is divided by 1 + u′v/c², which is larger than 1, so the result never exceeds c.
!Common mistakeChoosing the minus sign in the denominator would make the result LARGER than the Galilean sum and could exceed c.
26Short answer · ★ Challenge
Use the constancy of the speed of light to explain why two events that are simultaneous for one observer may not be simultaneous for another observer moving relative to the first.
Show answer
Model answer: Each observer judges when events happened from when the light arrives and its speed, which is c for both. If the events are equally far from the first observer and her light signals arrive together, she says they were simultaneous. The second observer moves towards one event and away from the other in her frame; using speed c in his own frame he concludes the light from one set off earlier. So the two disagree.
!Common mistakeSaying "one of them sees the light late because light is slower for him" is wrong; both use the same speed c.
27Fill in the blank
At v = 0.95c the relativistic mass of a proton is about ______ times its rest mass.
Show answer
Answer: 3.2
γ = 1/√(1 − 0.95²) = 1/√0.0975 = 3.2.
!Common mistakeAnswering 1.05 or 0.95 uses v/c itself; you must work out γ.
28Short answer · ★ Challenge
Using p = γm₀v, explain why a constant force acting for a long time on a particle can never make it reach the speed of light.
Show answer
Model answer: A constant force keeps increasing the momentum (F = Δp/Δt) without limit. But as v approaches c, γ grows without limit, so a huge increase in p produces only a tiny increase in v. The speed creeps closer and closer to c but never reaches it, because p would have to be infinite.
!Common mistakeUsing F = ma with constant m predicts the speed passes c; at high speed the momentum, not the speed, keeps growing.
29Short answer · ★ Challenge
A learner is in a closed lift with no windows, moving upward at a constant velocity. She drops a ball. Can she tell from the ball's motion whether the lift is moving? Explain using the first postulate.
Show answer
Model answer: No. The ball falls straight down with the same acceleration as in a lift at rest. The lift is an inertial frame, and by the first postulate the laws of physics are the same in all inertial frames, so no experiment inside can detect uniform motion.
!Common mistakeThinking the ball would land behind or move more slowly confuses constant velocity with acceleration.
30Multiple choice
In 1971, atomic clocks flown east and west around the world came back showing different times from identical clocks left on the ground. This experiment gave evidence for
- Athe ether wind of the Michelson–Morley test
- Blength contraction of the aircraft wings
- Cthe photoelectric effect inside the clocks
- Dtime dilation, as predicted by relativity
Show answer
Answer: D. time dilation, as predicted by relativity
The flown clocks lost or gained time by the amounts relativity predicts for their speed (and height).
!Common mistakeChoosing the ether wind is wrong: the experiment measured time, and the ether had already been abandoned.
31Short answer · ★ Challenge
A 2.0 m tall astronaut first lies along the direction of motion and later stands at right angles to it, while her ship moves at 0.80c relative to Earth. What height do Earth observers measure in each case?
Show answer
Model answer: γ = 1.67. Lying along the motion: L = 2.0 ÷ 1.67 = 1.2 m. Standing at right angles to the motion: 2.0 m, since lengths perpendicular to the motion are not contracted.
!Common mistakeContracting both measurements is the common error; only the dimension along the motion is shortened.
32Fill in the blank
A spaceship moves with γ = 3. One hour on the ship's clock is measured from Earth to last ______ hours.
Show answer
Answer: 3
Δt = γΔt₀ = 3 × 1 h = 3 h.
!Common mistakeAnswering ⅓ hour reverses time dilation: the moving clock is seen to run SLOW, so the interval is longer.
33Multiple choice · ★ Challenge
An astronaut's pulse is 72 beats per minute measured on her own ship. The ship passes Earth at 0.60c. What pulse rate do Earth observers work out for her?
- A90 beats per minute
- B72 beats per minute
- C58 beats per minute
- D43 beats per minute
Show answer
Answer: C. 58 beats per minute
γ = 1.25; each beat takes 1.25 times longer as seen from Earth, so the rate is 72 ÷ 1.25 ≈ 58 per minute.
!Common mistakeChoosing 90 multiplies the RATE by γ; it is the time between beats that is multiplied by γ, so the rate falls.
34Short answer
Give two pieces of experimental evidence that time dilation really happens.
Show answer
Model answer: Any two: muons made high in the atmosphere reach the ground in large numbers although their proper lifetime is too short; atomic clocks flown in aircraft come back showing a different time from identical clocks left on the ground; fast unstable particles in accelerators live longer in the laboratory frame; GPS satellite clocks must be corrected for their speed.
!Common mistakeQuoting the twin paradox as "evidence" is wrong; it is a thought experiment, not a measurement.
35Short answer · ★ Challenge
Calculate γ for v = 0.10c, 0.50c, 0.90c and 0.99c, and use your results to explain when relativistic effects can be ignored.
Show answer
Model answer: γ ≈ 1.005, 1.15, 2.29 and 7.09. Below about 0.1c, γ differs from 1 by less than 1%, so Newtonian physics is accurate; effects grow quickly above 0.5c and become huge as v approaches c.
!Common mistakeExpecting γ to rise steadily (doubling at 0.5c) ignores that it stays near 1 until v is a large fraction of c.
36True or false
Unstable particles moving close to c in accelerators such as CERN are measured to live longer in the laboratory than identical particles at rest, as special relativity predicts.
Show answer
Answer: True
Their lifetime in the lab is γ times their proper lifetime.
!Common mistakeSome think the particles become more stable; in their own frame they decay normally, the lab sees their clocks dilated.
37Multiple choice · ★ Challenge
In a particle accelerator, electrons of rest mass 9.11 × 10⁻³¹ kg travel at 0.80c. Use p = γm₀v to find the momentum of each. (c = 3.0 × 10⁸ m/s)
- A2.2 × 10⁻²² kg m/s
- B1.3 × 10⁻²² kg m/s
- C4.6 × 10⁻²² kg m/s
- D3.6 × 10⁻²² kg m/s
Show answer
Answer: D. 3.6 × 10⁻²² kg m/s
γ = 1.67, v = 2.4 × 10⁸ m/s: p = 1.67 × 9.11 × 10⁻³¹ × 2.4 × 10⁸ ≈ 3.6 × 10⁻²² kg m/s.
!Common mistakeChoosing 2.2 × 10⁻²² kg m/s is the classical m₀v, which is too small at 0.80c; 1.3 × 10⁻²² divides by γ instead of multiplying.
38Multiple choice
Cosmic-ray protons reach a detector at 0.28 of the speed of light. By what factor γ are their clocks dilated?
- A0.96
- B1.28
- C1.04
- D1.09
Show answer
Answer: C. 1.04
γ = 1/√(1 − 0.28²) = 1/√0.9216 = 1/0.96 = 1.04.
!Common mistakeChoosing 0.96 forgets to take the reciprocal; γ is always 1 or more.
39Multiple choice · ★ Challenge
One twin travels at 0.60c to a star 3.0 light-years away and straight back, while the other stays on Earth, who ages 10 years. How much does the travelling twin age?
- A12.5 years
- B8.0 years
- C10 years
- D6.0 years
Show answer
Answer: B. 8.0 years
Earth time = 6.0 light-years ÷ 0.60c = 10 years; γ = 1.25, so the traveller ages 10 ÷ 1.25 = 8.0 years.
!Common mistakeChoosing 12.5 years multiplies by γ; the traveller's clock runs slow as seen from Earth, so she ages LESS.
40Multiple choice
A spaceship moving at 0.90c relative to Earth shines a torch forwards. Using the relativistic addition formula with u′ = c, what speed of light does Earth measure?
- A1.9c
- B0.10c
- Cc
- D0.90c
Show answer
Answer: C. c
u = (c + 0.90c) ÷ (1 + 0.90) = 1.90c ÷ 1.90 = c.
!Common mistakeChoosing 1.9c adds the speeds as Galileo would; the formula gives exactly c, matching the second postulate.
41Short answer · ★ Challenge
Show that the Galilean velocity rule predicts that light from the headlamp of a bus moving at 30 m/s travels at c + 30 m/s relative to the road, and explain why this disagrees with experiment.
Show answer
Model answer: Galilean addition: u = u′ + v = c + 30 m/s. But experiments (Michelson–Morley, light from fast-moving sources) always find light travels at exactly c in vacuum for every observer, whatever the motion of the source. The Galilean rule fails for light and has to be replaced by the relativistic one.
!Common mistakeSaying "30 m/s is too small to matter" misses the point: measurements show NO change at all, even with sources moving close to c.
42Multiple choice
A moto at 12 m/s overtakes a cyclist riding at 5.0 m/s in the same direction. Using Galilean relativity, what is the speed of the moto relative to the cyclist?
- A7.0 m/s
- B17 m/s
- C12 m/s
- D2.4 m/s
Show answer
Answer: A. 7.0 m/s
Relative speed = 12 − 5.0 = 7.0 m/s (same direction, so subtract).
!Common mistakeChoosing 17 m/s adds the speeds, which is right only when they move towards each other.
43Short answer · ★ Challenge
A car battery stores 2.0 MJ of chemical energy when fully charged. By how much is its mass greater than when it is flat, and why is this never noticed? (c = 3.0 × 10⁸ m/s)
Show answer
Model answer: Δm = E/c² = 2.0 × 10⁶ ÷ 9.0 × 10¹⁶ ≈ 2.2 × 10⁻¹¹ kg. This is about a millionth of a millionth of a 15 kg battery, far too small for any balance to detect.
!Common mistakeUsing c instead of c² gives 6.7 × 10⁻³ kg, which would be easy to weigh; remember to square c.
44Multiple choice · ★ Challenge
A pot holding 2.0 kg of water is heated from 20 °C to 100 °C (specific heat capacity 4200 J kg⁻¹ K⁻¹). By how much does the mass of the water increase? (c = 3.0 × 10⁸ m/s)
- A2.2 × 10⁻³ kg
- B7.5 × 10⁻¹² kg
- C6.0 × 10²² kg
- D7.5 × 10⁻⁹ kg
Show answer
Answer: B. 7.5 × 10⁻¹² kg
Q = mcΔθ = 2.0 × 4200 × 80 = 6.72 × 10⁵ J; Δm = Q/c² = 6.72 × 10⁵ ÷ 9.0 × 10¹⁶ ≈ 7.5 × 10⁻¹² kg.
!Common mistakeChoosing 2.2 × 10⁻³ kg divides by c instead of c²; 7.5 × 10⁻⁹ kg comes from a slip with kJ and J.
45Multiple choice
A football pitch is 100 m long. A spaceship flies along its length at 0.60c. What length does the crew measure for the pitch?
- A125 m
- B100 m
- C60 m
- D80 m
Show answer
Answer: D. 80 m
The pitch moves relative to the crew, so it is contracted: L = L₀/γ = 100 ÷ 1.25 = 80 m.
!Common mistakeChoosing 125 m multiplies by γ; moving lengths get SHORTER. Choosing 100 m forgets that the pitch moves in the crew's frame.
46Short answer · ★ Challenge
A passenger walks forwards at 1.5 m/s in a bus moving at 20 m/s. Show that the relativistic velocity formula gives practically the same answer as Galilean addition. (c = 3.0 × 10⁸ m/s)
Show answer
Model answer: The denominator is 1 + u′v/c² = 1 + (1.5 × 20) ÷ (9.0 × 10¹⁶) = 1 + 3.3 × 10⁻¹⁶. So u = 21.5 m/s divided by a number that differs from 1 by about 3 × 10⁻¹⁶: the answer is 21.5 m/s, the same as Galilean addition, to far more digits than can be measured.
!Common mistakeForgetting to square c gives a correction of about 10⁻⁷, which is still small but wrong by a factor of c.
47Short answer
A spaceship moving at 0.50c relative to Earth fires one laser beam forwards and one backwards. What speed does a crew member measure for each beam, and what speed does an observer on Earth measure for each? Which postulate is this?
Show answer
Model answer: The crew measure c for both beams, and the Earth observer also measures c for both (not 1.5c or 0.5c). This is the second postulate: the speed of light in vacuum is the same for all observers, whatever the motion of the source or observer.
!Common mistakeAdding and subtracting 0.5c to get 1.5c and 0.5c uses Galilean addition, which does not work for light.
48Short answer · ★ Challenge
A lamp on a space station flashes every 2.0 s by the station clock. A ship passes the station at 0.80c. Find the time between flashes measured on the ship, and state which observer measures the proper time.
Show answer
Model answer: γ = 1/√(1 − 0.64) = 1/0.60 = 1.67. On the ship, Δt = γΔt₀ = 1.67 × 2.0 ≈ 3.3 s. The station measures the proper time, because the flashes happen at the same place (the lamp) in the station's frame.
!Common mistakeSome say the ship measures the proper time because "it is moving"; proper time belongs to the frame in which both events happen at the same place.
49Multiple choice
What is the rest energy of a proton (mass 1.67 × 10⁻²⁷ kg) in MeV? (c = 3.0 × 10⁸ m/s, 1 MeV = 1.6 × 10⁻¹³ J)
- A0.511 MeV
- B1.5 × 10⁻¹⁰ MeV
- C9.39 MeV
- D939 MeV
Show answer
Answer: D. 939 MeV
E = mc² = 1.67 × 10⁻²⁷ × 9.0 × 10¹⁶ = 1.50 × 10⁻¹⁰ J; ÷ 1.6 × 10⁻¹³ = 939 MeV.
!Common mistakeChoosing 1.5 × 10⁻¹⁰ MeV forgets to divide by 1.6 × 10⁻¹³ (that number is the energy in joules); 0.511 MeV is the electron's rest energy.
50Short answer · ★ Challenge
An electron (m₀ = 9.11 × 10⁻³¹ kg) moves at 0.90c. Calculate its kinetic energy using the relativistic formula and using ½m₀v², and comment on the difference. (c = 3.0 × 10⁸ m/s)
Show answer
Model answer: m₀c² = 8.20 × 10⁻¹⁴ J, γ = 2.29: KE = (γ − 1)m₀c² = 1.29 × 8.20 × 10⁻¹⁴ ≈ 1.06 × 10⁻¹³ J. Classical: ½ × 9.11 × 10⁻³¹ × (2.7 × 10⁸)² ≈ 3.3 × 10⁻¹⁴ J. The true KE is about 3.2 times the classical value, so ½mv² is badly wrong at this speed.
!Common mistakeUsing γm₀c² as the kinetic energy forgets to subtract the rest energy.