Q: Which equation correctly expresses time dilation for an object moving at speed $$v$$ relative to an observer?

$$\Delta t = \frac{\Delta \tau}{\sqrt{1 - v^2$/c^2$$}}$

Q: A spaceship travels at $$0.8c$$ relative to Earth. If its proper length is $$100\ \mathrm{m}$$, what is its length as measured by an observer on Earth?

$$L = 100\ \mathrm{m} \times \sqrt{1 - (0.8)^2$}$$

Q: A 100 g ball moves at $$100\ \mathrm{m/s}. $$What is its rest energy?

$$E_0 = (0.10$$\ \mathrm{kg}) \times (3.00 \times $$10^8$$\ \mathrm{m/s}$$)^2$$

Question 1

Which of the following statements best describes an inertial reference frame?

Accepted Answer

A reference frame in which Newton’s first law is valid and which moves at constant velocity relative to other inertial frames.

Question 2

According to Einstein’s principle of relativity, which of the following is true?

Accepted Answer

The speed of light is the same in all inertial reference frames.

Question 3

A firecracker explodes at position $$x = 900\ \mathrm{m} at$$ $$t = 4.0\ \mu\mathrm{s}$$, as seen by an observer at the origin. If light travels at $$300\ \mathrm{m}/\mu\mathrm{s}$$, when did the explosion actually occur?

Accepted Answer

$$t = 1.0\ \mu\mathrm{s}$$

Question 4

Which equation correctly expresses time dilation for an object moving at speed $$v$$ relative to an observer?

Accepted Answer

$$\Delta t = \frac{\Delta 	au}{\sqrt{1 - v^2$/c^2$$}}$

Question 5

A muon is created in the upper atmosphere and travels toward Earth at $$0.998c. $$Its proper lifetime is $$2.2\ \mu\mathrm{s}. $$How long does it live as measured by an observer on Earth?

Accepted Answer

$$\Delta t = \frac{2.2\ \mu\mathrm{s}}{\sqrt{1 - (0.998)^2$}}$$

Question 6

Which of the following best describes length contraction?

Accepted Answer

The length of an object is less when it moves relative to the observer.

Question 7

A spaceship travels at $$0.8c$$ relative to Earth. If its proper length is $$100\ \mathrm{m}$$, what is its length as measured by an observer on Earth?

Accepted Answer

$$L = 100\ \mathrm{m} 	imes \sqrt{1 - (0.8)^2$}$$

Question 8

Which of the following is a consequence of the relativity of simultaneity?

Accepted Answer

Events that are simultaneous in one inertial frame may not be simultaneous in another.

Question 9

The Lorentz transformation for the $$x-$$coordinate between two inertial frames S and S′ moving at velocity $$v$$ relative to each other is:

Accepted Answer

$$x' = \gamma (x - vt)$$

Question 10

A rocket moves at $$0.9c$$ relative to Earth. It fires a probe forward at $$0.8c$$ relative to the rocket. What is the probe’s speed relative to Earth?

Accepted Answer

$$u = \frac{0.8c + 0.9c}{1 + (0.8)(0.9)}$$

Question 11

Which of the following is an invariant quantity in special relativity?

Accepted Answer

The spacetime interval $$s^2$$ = $$c^2$$ \Delta $$t^2$$ - \Delta $$x^2$$

Question 12

Which equation gives the relativistic momentum of a particle with mass $$m$$ and velocity $$u$$?

Accepted Answer

$$p = \gamma_p m u$$

Question 13

A 100 g ball moves at $$100\ \mathrm{m/s}. $$What is its rest energy?

Accepted Answer

$$E_0 = (0.10$$\ \mathrm{kg}) 	imes (3.00 	imes $$10^8$$\ \mathrm{m/s}$$)^2$$

Question 14

Which of the following best describes the outcome of electron-positron annihilation?

Accepted Answer

Two photons are emitted in opposite directions, each with energy approximately $$m_e c^2$

Question 15

Which of the following is a real-world application that requires relativistic corrections?

Accepted Answer

Global Positioning System (GPS)

Question 16

In the twin paradox, why is the twin who travels on the starship younger upon return?

Accepted Answer

Because the traveling twin’s clock runs slower due to time dilation, and only the Earth twin remains in an inertial frame for the entire journey.

Question 17

Which equation expresses the total relativistic energy of a particle?

Accepted Answer

$$E = \gamma_p m c^2$

Question 18

A uranium-235 nucleus undergoes fission and loses mass $$\Delta m. $$How much energy is released?

Accepted Answer

$$E = \Delta m c^2$

Question 19

Which of the following statements is true regarding the cosmic speed limit?

Accepted Answer

No particle, wave, or information can travel faster than the speed of light $$c.$$

Chapter 36: Special Relativity – Concepts, Transformations, and Applications

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