Relativistic Momentum Calculations in Physics

1. Relativistic momentum is:

P = mv

P = γmv

P = mγ/v

P = γm/v²

Concept: relativistic momentum formula. γ corrects momentum at high speeds. It becomes important when v is a noticeable fraction of c.

Explanation

Concept: relativistic momentum formula. γ corrects momentum at high speeds. It becomes important when v is a noticeable fraction of c.

2. If v = 0.6c, then γ is greater than 1.

True

False

Concept: gamma behavior. Any nonzero v gives γ>1. At 0.6c, γ is noticeably above 1, so relativistic corrections matter.

Explanation

Concept: gamma behavior. Any nonzero v gives γ>1. At 0.6c, γ is noticeably above 1, so relativistic corrections matter.

3. A particle has m = 2 kg and v = 0.8c. If γ is given as 1.67 (approx), p is closest to:

1.6c kg·m/s

2.67c kg·m/s

3.34c kg·m/s

0.96c kg·m/s

Concept: plugging in values. p = γmv ≈ 1.67 × 2 × 0.8c = 2.67c (kg·m/s). Using c as a symbol avoids messy decimals.

Explanation

Concept: plugging in values. p = γmv ≈ 1.67 × 2 × 0.8c = 2.67c (kg·m/s). Using c as a symbol avoids messy decimals.

4. If γ doubles while m and v stay the same, momentum p ______.

Concept: direct proportionality to γ. p is directly proportional to γ when m and v are fixed. This highlights how strongly γ controls p at high speeds.

Explanation

Concept: direct proportionality to γ. p is directly proportional to γ when m and v are fixed. This highlights how strongly γ controls p at high speeds.

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5. For the same object, which grows faster with speed near c:

Classical mv

Relativistic γmv

They grow at the same rate

Neither grows

Concept: nonlinear growth. γ increases sharply as v→c, so γmv grows faster than mv. This is a key high-speed difference.

Explanation

Concept: nonlinear growth. γ increases sharply as v→c, so γmv grows faster than mv. This is a key high-speed difference.

6. Relativistic momentum can be much larger than mv even if v changes only slightly near c.

True

False

Concept: sensitivity near c. Near c, γ is very sensitive to v. A small speed increase can produce a large momentum increase.

Explanation

Concept: sensitivity near c. Near c, γ is very sensitive to v. A small speed increase can produce a large momentum increase.

7. Which expression best shows why momentum becomes huge near c?

γ = 1 − v²/c²

γ = 1/√(1 − v²/c²)

γ = v/c

γ = c/v

Concept: gamma blow-up. As v approaches c, the denominator approaches zero. That makes γ (and thus momentum) grow without bound.

Explanation

Concept: gamma blow-up. As v approaches c, the denominator approaches zero. That makes γ (and thus momentum) grow without bound.

8. If v is very small, then γ ≈ 1 and p ≈:

γm/c

Mv

Mc²

M/v

Concept: classical limit. For small v, γ≈1, so p≈mv. This confirms that classical mechanics is the low-speed approximation.

Explanation

Concept: classical limit. For small v, γ≈1, so p≈mv. This confirms that classical mechanics is the low-speed approximation.

9. Relativistic momentum has the same units as classical momentum.

True

False

Concept: units consistency. γ is dimensionless, so it doesn’t change units. Momentum remains kg·m/s.

Explanation

Concept: units consistency. γ is dimensionless, so it doesn’t change units. Momentum remains kg·m/s.

10. If two identical masses move in opposite directions with equal speed, the total momentum is:

Twice one momentum

Zero (they cancel)

Negative infinity

Always positive

Concept: vector cancellation. Momentum is a vector. Equal magnitudes in opposite directions sum to zero in that frame.

Explanation

Concept: vector cancellation. Momentum is a vector. Equal magnitudes in opposite directions sum to zero in that frame.

11. The frame where total momentum is zero is often called the ______-of-momentum frame.

Concept: center-of-momentum frame. This frame is useful in collision analysis. It simplifies conservation equations.

Explanation

Concept: center-of-momentum frame. This frame is useful in collision analysis. It simplifies conservation equations.

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12. Which statement is correct?

Relativistic momentum decreases as v increases

Relativistic momentum increases as v increases

Relativistic momentum is independent of v

Relativistic momentum depends only on temperature

Concept: dependence on speed. Both v and γ contribute to momentum. Increasing speed increases momentum.

Explanation

Concept: dependence on speed. Both v and γ contribute to momentum. Increasing speed increases momentum.

13. In an isolated system, total momentum is conserved in relativity.

True

False

Concept: conservation. Conservation laws still apply. You just need the correct relativistic expressions.

Explanation

Concept: conservation. Conservation laws still apply. You just need the correct relativistic expressions.

14. A major reason SR changes momentum is that:

Time is absolute

Space and time transform via Lorentz transformations, changing how velocity-related quantities behave

Gravity is always present

Mass is always zero

Concept: Lorentz transformations. Momentum must transform consistently with spacetime. Lorentz transformations lead naturally to γ factors.

Explanation

Concept: Lorentz transformations. Momentum must transform consistently with spacetime. Lorentz transformations lead naturally to γ factors.

15. Which are true?

γ ≥ 1 for v < c

P = γmv

At v ≪ c, p ≈ mv

γ is negative near c

Concept: gamma facts. γ is never negative for real v<c. These are key checks students should remember.

Explanation

Concept: gamma facts. γ is never negative for real v<c. These are key checks students should remember.

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16. The expression p = γmv implies that momentum can grow without bound as v→c.

True

False

Concept: asymptotic behavior. γ diverges as v approaches c. That drives momentum toward infinity in the model, reinforcing the speed limit.

Explanation

Concept: asymptotic behavior. γ diverges as v approaches c. That drives momentum toward infinity in the model, reinforcing the speed limit.

17. If γ = 1.25, m = 4 kg, v = 10 m/s, then p is:

40 kg·m/s

50 kg·m/s

12.5 kg·m/s

5 kg·m/s

Concept: compute p. p = 1.25 × 4 × 10 = 50 kg·m/s. This shows a modest relativistic correction.

Explanation

Concept: compute p. p = 1.25 × 4 × 10 = 50 kg·m/s. This shows a modest relativistic correction.

18. Which is the best description of why relativistic momentum is used?

It is simpler

It matches experiments and keeps conservation laws valid at high speeds

It removes the need for units

It applies only to sound waves

Concept: evidence and consistency. High-speed experiments require relativistic formulas. They preserve conservation principles across frames.

Explanation

Concept: evidence and consistency. High-speed experiments require relativistic formulas. They preserve conservation principles across frames.

19. If you incorrectly use mv at 0.99c, you can get large errors.

True

False

Concept: large relativistic corrections. At 0.99c, γ is large. Ignoring γ leads to big underestimates of momentum.

Explanation

Concept: large relativistic corrections. At 0.99c, γ is large. Ignoring γ leads to big underestimates of momentum.

20. Grade 11 wrap-up (less obvious): the 'extra factor' in relativistic momentum ultimately comes from:

Friction

How simultaneity and time intervals differ between inertial frames

Temperature dependence of mass

Gravity waves

Concept: spacetime geometry in SR. Lorentz transformations reflect how time and simultaneity depend on the frame. γ encodes those differences, so it appears in momentum.

Explanation

Concept: spacetime geometry in SR. Lorentz transformations reflect how time and simultaneity depend on the frame. γ encodes those differences, so it appears in momentum.