Question 1

Suppose Alice is at rest in the ether and Bob moves to the right at speed $$v = \frac{3}{5}c. If $$Alice waits until her clock reads $$ct_A = 10\ \mathrm{m}$$, how far has Bob moved in Alice's frame during this time?

Accepted Answer

$$6\ \mathrm{m}$$

Question 2

Given Bob's speed $$v = \frac{3}{5}c$$, what is the Lorentz factor $$\gamma$$ for this scenario?

Accepted Answer

$$\gamma = \frac{5}{4}$$

Question 3

Using the time dilation formula, what is the reading on Bob's clock when Alice's clock reads $$ct_A = 10\ \mathrm{m}$$?

Accepted Answer

$$8\ \mathrm{m}$$

Question 4

Which of the following best describes 'proper time' in the context of special relativity?

Accepted Answer

The time interval measured by a single clock present at both events.

Question 5

If Bob waits until his clock reads $$ct_B = 8\ \mathrm{m}$$, how much time has elapsed on Alice's clock according to the time dilation formula?

Accepted Answer

$$6.4\ \mathrm{m}$$

Question 6

In the scenario where Bob and Bob' are standing on a platform of proper length $$L = 4.8\ \mathrm{m}$$, what length does Alice measure for the platform as it moves past her at $$v = \frac{3}{5}c$$?

Accepted Answer

$$3.84\ \mathrm{m}$$

Question 7

Which formula correctly expresses the relationship for length contraction in special relativity?

Accepted Answer

$$\ell = \frac{L}{\gamma}$$

Question 8

Why does Alice require a twin, Alice', at $$x_A = 6\ \mathrm{m} to $$directly compare her clock with Bob's at $$ct_A = 10\ \mathrm{m}$$?

Accepted Answer

Because Bob is not at Alice's location when $$ct_A = 10\ \mathrm{m}$$, so Alice' can observe Bob's clock without accounting for light travel time.

Question 9

In Minkowski geometry, what is the invariant interval $$s^2$$ between two time-like separated events with coordinate differences $$\Delta x$$ and $$\Delta (ct)$$?

Accepted Answer

$$s^2$$ = (\Delta $$(ct))^2$$ - (\Delta $$x)^2$$

Question 10

If Alice measures the length of Bob's moving platform by touching both ends simultaneously in her frame, what is the reading on Bob's clock when Alice's right hand touches Bob (the right end)?

Accepted Answer

$$5.12\ \mathrm{m}$$

Question 11

Which concept explains why Alice and Bob disagree on the simultaneity of events when measuring the length of the moving platform?

Accepted Answer

Relativity of simultaneity

Question 12

From Bob's perspective, what is the distance between Bob and Bob' (the proper length of the platform) if Alice sees the platform moving at $$v = \frac{3}{5}c$$ and measures the contracted length $$\ell = 3.84\ \mathrm{m}$$?

Accepted Answer

$$4.8\ \mathrm{m}$$

Question 13

If Alice holds her hands $$\ell = 3.84\ \mathrm{m}$$ apart to measure the moving platform, what distance do Bob and Bob' measure between her hands due to length contraction?

Accepted Answer

$$3.072\ \mathrm{m}$$

Question 14

Which of the following statements about time dilation is correct according to the principle of relativity?

Accepted Answer

Both Alice and Bob see the other's moving clock as running slow.

Question 15

If Bob's time axis in Alice's spacetime diagram has a slope of $$\frac{c}{v} = \frac{5}{3}$$, what is the slope of Bob's space axis in Minkowski geometry?

Accepted Answer

$$\frac{v}{c} = \frac{3}{5}$$

Question 16

Why is length contraction not considered a physical compression of the moving object?

Accepted Answer

Because it results from the relativity of simultaneity, not from any physical force acting on the object.

Question 17

A train of proper length $$L$$ moves at speed $$v$$ relative to an observer. The observer measures the train's length as $$\ell. $$Which equation relates $$\ell$$ and $$L$$?

Accepted Answer

$$\ell = L \sqrt{1 - \frac{v^2$}{c^2$}}$$

Question 18

If Alice and Bob synchronize their clocks using light signals, what principle ensures that both agree on the speed of light?

Accepted Answer

The invariance of the speed of light in all inertial frames

Question 19

Suppose Alice measures the time interval between two events using two synchronized clocks at different locations, while Bob measures the interval using a single clock he carries with him. Which observer measures the proper time?

Accepted Answer

Bob

Question 20

Which of the following best explains why the relativity of simultaneity is necessary for the reciprocity of time dilation?

Accepted Answer

Because without it, both observers could not consistently see the other's clock as running slow.

Special Relativity: Time Dilation, Length Contraction, and Spacetime Diagrams

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