Question 1

Alice bungee-jumps from a bridge above a deep gorge. Bob watches from the bridge. Let event $D$ be Alice's departure from Bob's location on the bridge, and event $R$ be her return to Bob's location on the bridge. Carol observes these events from a a train passing over the bridge, and uses synchronized clocks on the train to measure the time between Alice's departure and return.
-(c) Which person's watch or clocks register(s) a coordinate time between those events in some inertial frame?
A) Alice
B) Bob
C) Carol
D) Alice and Bob
E) Bob and Carol
F) Alice and Carol
G) All three observers

Accepted Answer

Bob and Carol are in inertial frames relative to the events of Alice's departure and return, so their clocks can measure the coordinate time between these events. Alice, however, is in a non-inertial frame due to the acceleration involved in bungee jumping, making her perspective unsuitable for measuring coordinate time in an inertial frame.

Question 2

The space time interval $\Delta s$ between two events can never be larger than the coordinate time $\Delta t$ between those events as measured in any inertial reference frame.

Accepted Answer

The space-time interval is a measure of the distance between two events in space-time, while the coordinate time is a measure of the time between those events as measured in a particular inertial reference frame. The space-time interval is always the same for all inertial reference frames, while the coordinate time can vary depending on the reference frame. Therefore, the space-time interval can never be larger than the coordinate time.

Question 3

Two events occur $5.0 \mathrm{~s}$ apart in time and $3.0 \mathrm{~s}$ apart in space. A clock traveling at a speed of $0.60 \mathrm{can}$ be present at both these events. What time interval will such a clock measure between the events?
A) $8.0 \mathrm{~s}$
B) $5.8 \mathrm{~s}$
C) $5.0 \mathrm{~s}$
D) $4.0 \mathrm{~s}$
E) $2.0 \mathrm{~s}$
F) Other (specify)

Accepted Answer

The time interval measured by a clock moving with the events is the proper time, calculated using the spacetime interval: $\Delta \tau = \sqrt{(\Delta t)^2 - (\Delta x)^2}$. Here, $\Delta t = 5.0 \, \mathrm{s}$ and $\Delta x = 3.0 \, \mathrm{s}$, so $\Delta \tau = \sqrt{5.0^2 - 3.0^2} = \sqrt{25 - 9} = \sqrt{16} = 4.0 \, \mathrm{s}$.

Question 4

Consider the events $A, B, C$, and $D$ shown in the space time diagram below.
 
-(a) What is the space time interval between events A and B?
A) $0 \mathrm{~s}$
B) $2 \mathrm{~s}$
C) $3 \mathrm{~s}$
D) $4 \mathrm{~s}$
E) $5 \mathrm{~s}$
F) Other (specify)

Accepted Answer

The answer of Consider the events $A, B, C$, and...

Question 5

Consider the events $A, B, C$, and $D$ shown in the space time diagram below.
 
-(b) Between A and C?
A) $0 \mathrm{~s}$
B) $2 \mathrm{~s}$
C) $3 \mathrm{~s}$
D) $4 \mathrm{~s}$
E) $5 \mathrm{~s}$
F) Other (specify)

Accepted Answer

The answer of Consider the events $A, B, C$, and...

Question 6

Consider the events $A, B, C$, and $D$ shown in the space time diagram below.
 
-(c) Between $A$ and $D$ ?
A) $0 \mathrm{~s}$
B) $2 \mathrm{~s}$
C) $3 \mathrm{~s}$
D) $4 \mathrm{~s}$
E) $5 \mathrm{~s}$
F) Other (specify)

Accepted Answer

The answer of Consider the events $A, B, C$, and...

Question 7

Consider the space time diagram below. Let the space time interval between events $O$ and $A$ be $\Delta s_{O A}$, and let the space time interval between events $O$ and $B$ be $\Delta s_{O B}$ Which of these two space time intervals is larger? (Assume that the $y$ and $z$ coordinates of all these events are zero.)

A) $\Delta s_{O A}>\Delta s_{O B}$
B) $\Delta s_{O A}<\Delta s_{O B}$
C) $\Delta s_{O A}=\Delta s_{O B}$
D) There is no way to tell from this diagram.

Accepted Answer

The answer of Consider the space time diagram below. Let...

Question 8

An inertial clock present at two events always measures a shorter time than a pair of synchronized clocks in any inertial reference frame would register between the same two events (as long as the events don't occur at the same place in that frame).

Accepted Answer

According to the theory of relativity, an inertial clock moving between two events measures the proper time, which is the shortest time interval compared to any other clocks in different inertial frames that are not at rest relative to the events.

Question 9

Consider a train moving at a speed of 0.6 relative to the ground. A light in one of its windows blinks repeatedly. An observer on the ground will necessarily see (not observe) those blinks to be separated by a larger time interval than a person on the train would.

Accepted Answer

The speed of light is constant for all observers, regardless of their relative motion, according to the theory of relativity. Therefore, the time interval between the light blinks, as seen (not observed, which might imply relativistic effects like time dilation) directly, would not be affected by the train's motion relative to the observer.

Question 10

Suppose we carefully synchronize two identical atomic clocks initially standing next to each other (call them $A$ and $B$ ). We put clock $B$ on a jet plane, which then flies around the world at an essentially constant speed of $300 \mathrm{~m} / \mathrm{s}$, returning $134,000 \mathrm{~s}(37.1 \mathrm{~h})$ later. We then again compare the two clocks. Assume the earth's surface defines an inertial reference frame, and ignore the possible effects of gravity.
-(a) Which clock measures the space time interval between the synchronization and comparison events?
A) Clock $A$
B) Clock $B$
C) Both
D) Neither

Accepted Answer

Clock $A$ measures the proper time between the synchronization and comparison events because it remains at rest in the inertial reference frame defined by the earth's surface.

Question 11

Suppose we carefully synchronize two identical atomic clocks initially standing next to each other (call them $A$ and $B$ ). We put clock $B$ on a jet plane, which then flies around the world at an essentially constant speed of $300 \mathrm{~m} / \mathrm{s}$, returning $134,000 \mathrm{~s}(37.1 \mathrm{~h})$ later. We then again compare the two clocks. Assume the earth's surface defines an inertial reference frame, and ignore the possible effects of gravity.
-(b) Which clock measures a coordinate time between the synchronization and comparison events?
A) Clock $A$
B) Clock $B$
C) Both
D) Neither

Accepted Answer

Clock A measures the coordinate time between the synchronization and comparison events because it remains in the inertial reference frame defined by the Earth's surface, while clock B is moving and thus experiences time dilation according to the theory of relativity.

Question 12

Suppose we carefully synchronize two identical atomic clocks initially standing next to each other (call them $A$ and $B$ ). We put clock $B$ on a jet plane, which then flies around the world at an essentially constant speed of $300 \mathrm{~m} / \mathrm{s}$, returning $134,000 \mathrm{~s}(37.1 \mathrm{~h})$ later. We then again compare the two clocks. Assume the earth's surface defines an inertial reference frame, and ignore the possible effects of gravity.
-(c) Which clock measures the shorter time interval between the synchronization and comparison events (or do both measure the same time)?
A) Clock A
B) Clock $B$
C) Both
D) Neither

Accepted Answer

Clock B measures the shorter time interval due to time dilation effects as per the theory of relativity. Since clock B is moving at a significant speed relative to clock A, time for clock B slows down compared to clock A, which remains stationary on Earth.

Question 13

In the round-the-world experiment described in problem R4T.2, what is the minimum accuracy over the experiment's duration that the clocks must have to clearly display the relativistic effect?
A) Both clocks must be accurate to the nearest $10 \mathrm{~ms}$.
B) Both clocks must be accurate to the nearest $10 \mu \mathrm{s}$.
C) Both clocks must be accurate to the nearest $10 \mathrm{~ns}$.
D) Both clocks must be accurate to the nearest 10 ps.

Accepted Answer

The relativistic effect in the round-the-world experiment is very small, so clocks need to be accurate to the nearest 10 ns to detect the time difference caused by relativistic effects.

Question 14

Jennifer bungee-jumps from a bridge (event A). Jennifer's bungee cord is perfectly elastic, so she bounces exactly back up to the bridge and lands on her feet (event B). The time between these events is measured by Jennifer's watch, a stopwatch held by Jennifer's friend, Rob, who is standing on the bridge, and by two passengers (one present at event $A$ and one present at event $B$ ) who are riding on a train traveling at a constant velocity across the bridge at the time (the passengers have synchronized watches and compare readings later). Assuming the earth's frame is inertial, who measures
-(a) the longest time interval between these events?
A) Jennifer
B) Rob
C) The train passengers

Accepted Answer

The train passengers measure the longest time interval due to time dilation, as they are in a different inertial frame moving relative to the events.

Question 15

Jennifer bungee-jumps from a bridge (event A). Jennifer's bungee cord is perfectly elastic, so she bounces exactly back up to the bridge and lands on her feet (event B). The time between these events is measured by Jennifer's watch, a stopwatch held by Jennifer's friend, Rob, who is standing on the bridge, and by two passengers (one present at event $A$ and one present at event $B$ ) who are riding on a train traveling at a constant velocity across the bridge at the time (the passengers have synchronized watches and compare readings later). Assuming the earth's frame is inertial, who measures
-(b) the shortest time interval between these events?
A) Jennifer
B) Rob
C) The train passengers

Accepted Answer

In the scenario described, Jennifer is in a non-inertial frame of reference due to the acceleration experienced during the bungee jump. According to the theory of relativity, time dilation occurs for observers in motion relative to an inertial frame. However, since Jennifer is accelerating, her frame is non-inertial, and the effects of acceleration could cause her to measure a different time interval compared to observers in inertial frames. Rob and the train passengers are in inertial frames relative to the event, with the train passengers moving at a constant velocity and Rob presumably stationary relative to the Earth. Among these, the non-inertial observer (Jennifer) would experience the most significant effects on time measurement due to her unique state of motion, potentially measuring the shortest time interval due to the complexities introduced by acceleration and the equivalence principle.

Question 16

Jennifer bungee-jumps from a bridge (event A). Jennifer's bungee cord is perfectly elastic, so she bounces exactly back up to the bridge and lands on her feet (event B). The time between these events is measured by Jennifer's watch, a stopwatch held by Jennifer's friend, Rob, who is standing on the bridge, and by two passengers (one present at event $A$ and one present at event $B$ ) who are riding on a train traveling at a constant velocity across the bridge at the time (the passengers have synchronized watches and compare readings later). Assuming the earth's frame is inertial, who measures
-(c) the space time interval between the events?
A) Jennifer
B) Rob
C) The train passengers

Accepted Answer

The answer of Jennifer bungee-jumps from a bridge (event A)....

Question 17

In the situation described in problem R4T.4, the train passengers are moving, but Rob is at rest. Therefore, the train passengers measure less time between the events than Rob does.

Accepted Answer

In the situation described, which involves relative motion and the principles of special relativity, Rob, who is at rest relative to the event, would actually measure more time between the events (due to time dilation) than the train passengers who are in motion.

Question 18

Suppose we synchronize two atomic clocks at a point at $45^{\circ}$ south latitude, and then move one clock directly north to the earth's equator and the other directly south to the south pole, where they remain for some years in climate-controlled enclosures that keep them at the same temperature and humidity. We then reunite the clocks at the origin point and compare them again. Which (if either) has registered a shorter time between the synchronization and comparison events?
A) The clock at the equator}
B) The clock at the south pole
C) Both clocks read the same time.

Accepted Answer

The clock at the equator will register a shorter time due to the effects of general relativity, as it is further from the Earth's center and experiences less gravitational time dilation compared to the clock at the south pole.

Question 19

GPS satellites go around the earth in orbits that have a common radius of $26,600 \mathrm{~km}$ and a period of $12 \mathrm{~h}$. Roughly how much less time would an atomic clock on a GPS satellite register between two events separated by exactly 24 $\mathrm{h}$ than clocks in the reference frame of the earth (ignoring gravitational effects on the satellite's clock rate)?
A) About $10 \mathrm{~ms}$
B) About $1 \mathrm{~ms}$
C) About $100 \mu \mathrm{s}$
D) About 10 us
E) About 1 rs
F) About 10 ns

Accepted Answer

The answer of GPS satellites go around the earth in...

Question 20

The coordinate time between two given events is shortest in the inertial frame where their spatial separation is the smallest.

Accepted Answer

The coordinate time between two given events is shortest in the inertial frame where their spatial separation is the smallest because the speed of light is the same in all inertial frames, and the coordinate time between two events is the time it takes light to travel between them.

Quiz 3: The Laws of Physics Are Frame-Independent Relativity

The space time interval Δs\Delta sΔs between two events can never be larger than the coordinate time Δt\Delta tΔt between those events as measured in any inertial reference frame.

Two events occur 5.0 s5.0 \mathrm{~s}5.0 s apart in time and 3.0 s3.0 \mathrm{~s}3.0 s apart in space. A clock traveling at a speed of 0.60can0.60 \mathrm{can}0.60can be present at both these events. What time interval will such a clock measure between the events?

Consider the events A,B,CA, B, CA,B,C , and DDD shown in the space time diagram below. -(a) What is the space time interval between events A and B?

Consider the events A,B,CA, B, CA,B,C , and DDD shown in the space time diagram below. -(b) Between A and C?

Consider the events A,B,CA, B, CA,B,C , and DDD shown in the space time diagram below. -(c) Between AAA and DDD ?

An inertial clock present at two events always measures a shorter time than a pair of synchronized clocks in any inertial reference frame would register between the same two events (as long as the events don't occur at the same place in that frame).

Consider a train moving at a speed of 0.6 relative to the ground. A light in one of its windows blinks repeatedly. An observer on the ground will necessarily see (not observe) those blinks to be separated by a larger time interval than a person on the train would.

In the round-the-world experiment described in problem R4T.2, what is the minimum accuracy over the experiment's duration that the clocks must have to clearly display the relativistic effect?

In the situation described in problem R4T.4, the train passengers are moving, but Rob is at rest. Therefore, the train passengers measure less time between the events than Rob does.

The coordinate time between two given events is shortest in the inertial frame where their spatial separation is the smallest.

The space time interval $\Delta s$ between two events can never be larger than the coordinate time $\Delta t$ between those events as measured in any inertial reference frame.

Two events occur $5.0 \mathrm{~s}$ apart in time and $3.0 \mathrm{~s}$ apart in space. A clock traveling at a speed of $0.60 \mathrm{can}$ be present at both these events. What time interval will such a clock measure between the events?