Question 1

The astronaut whose heart rate on Earth is 60 beats/min increases his velocity to v = 0.60c.Now what is his heart rate as measured by an Earth observer?

Accepted Answer

D

Question 2

If astronauts could travel at v = 0.90c,we on Earth would say it takes (4.2/0.90)= 4.7 years to reach Alpha Centauri,4.2 lightyears away.The astronauts disagree.How much time passes on the astronaut's clocks?

Accepted Answer

A

Question 3

A muon formed high in the Earth's atmosphere travels at a speed 0.990c for a distance of 6.40 km before it decays.What is the muon's lifetime as measured in its reference frame?

Accepted Answer

D

Question 4

A space probe has a 24.7-m length when measured at rest.What length does an observer at rest measure when the probe is going by at a speed of 0.860c?

Accepted Answer

The answer of A space probe has a 24.7-m length...

Question 5

A ground observer measures the period of a pendulum moving as a part of an inertial frame of reference to be 2.30 s as the inertial frame moves by at a velocity of 0.800c.What would the observed period be of the same pendulum if its inertial frame were at rest with respect to the observer?

Accepted Answer

According to time dilation, the time interval between two events in a moving frame of reference is longer than that in a rest frame of reference. The observed period in the ground observer's frame will be longer than the actual period in the pendulum's rest frame. Using the time dilation equation:

Δt' = Δt/γ

where Δt is the time interval in the rest frame, Δt' is the time interval in the moving frame, and γ is the Lorentz factor:

γ = 1/√(1 - v^2/c^2)

where v is the velocity of the moving frame and c is the speed of light.

In this problem, Δt' = 2.30 s and v = 0.800c. Solving for γ:

γ = 1/√(1 - (0.800c)^2/c^2) = 1.67

Substituting into the time dilation equation:

Δt = Δt'/γ = 2.30 s/1.67 = 1.38 s

Therefore, the observed period of the pendulum in the rest frame of reference is 1.38 s, which corresponds to choice B.

Question 6

The observed relativistic length of a super rocket moving by the observer at 0.57c will be what factor times that of the measured rocket length if it were at rest?

Accepted Answer

The relativistic length contraction formula is L = L0 * sqrt(1 - v^2/c^2). For v = 0.57c, the factor is sqrt(1 - (0.57)^2) = 0.82.

Question 7

A spaceship is moving away from an observer at relativistic speed.Its length at rest is 100 m,but the observer measures its length as 67 m.A time interval of 100 s on the spaceship would be measured by the observer to be which of the following?

Accepted Answer

This problem can be solved using the formula for length contraction and time dilation.

Length contraction:

L' = L*(1 - v^2/c^2)^(-1/2)

where L is the length at rest, v is the velocity, c is the speed of light, and L' is the length as measured by the observer.

Substituting the given values, we get:

67 = 100*(1 - v^2/c^2)^(-1/2)

Solving for v/c, we get:

v/c = (1 - (67/100)^2)^(1/2) = 0.8

Time dilation:

t' = t*(1 - v^2/c^2)^(1/2)

where t is the time interval on the spaceship and t' is the time interval as measured by the observer.

Substituting the given values and the value of v/c that we just calculated, we get:

t' = 100*(1 - 0.8^2)^(1/2) = 60 s

Therefore, the best choice is B) 150 s.

Question 8

An astronaut is traveling at a high rate of speed from the Earth to a distant star system.Observers here on Earth note the relativistic effect that the astronaut's clocks run slow.What relativistic effect,if any,does the astronaut observe?

Accepted Answer

The astronaut observes length contraction, a relativistic effect where the distance between two points (in this case, from the Earth to the star system) appears shorter for an observer moving at a significant fraction of the speed of light relative to those points. This is a direct consequence of the principles of special relativity.

Question 9

A rocket ship is 74.0 m in length when measured before leaving the launching pad.What would its velocity be if a ground observer measured its length as 60.0 m while it is in flight? (c = 3.00 × 10⁸ m/s)

Accepted Answer

The length contraction formula is L = L₀√(1-v²/c²). Solving for v when L = 60.0 m and L₀ = 74.0 m gives v ≈ 1.76 × 10⁸ m/s.

Question 10

A muon formed high in Earth's atmosphere travels at a speed 0.9900c for a distance (as we see it)of 9200 m before it decays.How far does the muon travel as measured in its frame?

Accepted Answer

The distance measured in the muon's frame is contracted due to its high speed. Using the Lorentz contraction formula L = L0 * sqrt(1 - v^2/c^2), where L0 is the distance in Earth's frame (9200 m) and v = 0.9900c, the contracted distance L is approximately 1298 m.

Question 11

An astronaut at rest has a heart rate of 65 beats/min.What will her heart rate be as measured by an earth observer when the astronaut's spaceship goes by the earth at a speed of 0.80c?

Accepted Answer

We can use the time dilation formula, which states that the observed time interval is equal to the proper time divided by the Lorentz factor, to find the heart rate as measured by an earth observer. The Lorentz factor can be calculated as gamma = 1 / sqrt(1 - v^2/c^2), where v is the speed of the spaceship as a fraction of the speed of light (0.80c in this case) and c is the speed of light.

gamma = 1 / sqrt(1 - (0.80c)^2/c^2) 
gamma = 1.67

The observed heart rate can be found by dividing the proper heart rate (65 beats/min) by gamma:

observed heart rate = proper heart rate / gamma 
observed heart rate = 65 beats/min / 1.67 
observed heart rate ≈ 39 beats/min

Therefore, the astronaut's heart rate as measured by an earth observer will be 39 beats/min, which is closest to choice B.

Question 12

A knight on horseback holds a 10-m lance.The horse can run at 0.78c.(It wins most of its races!)How long will the lance appear to a person that is standing still on the ground as the horse runs past?

Accepted Answer

We can use the length contraction formula:
L' = L / sqrt(1 - v^2/c^2)
Where L = 10 m (length of the lance)
v = 0.78c (velocity of the horse)
c = speed of light
Plugging in the values, we get:
L' = 10 / sqrt(1 - 0.78^2) = 6.3 m
Therefore, the lance will appear to be 6.3 m long to a person standing still on the ground as the horse runs past.

Question 13

Muons at speed 0.9994c are sent round and round a circular storage ring of radius 1500 m.If a muon at rest decays into other particles after an average T = 2.2 × 10−⁶ s,how many trips around the storage ring do we expect the 0.9994c muons to make before they decay?

Accepted Answer

The time dilation formula is:$$t' = \frac{t}{\gamma}$$where $t'$ is the time measured in the moving frame, $t$ is the time measured in the stationary frame, and $\gamma$ is the Lorentz factor:$$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$In this problem, $t = 2.2 	imes 10^{-6} s$, $v = 0.9994c$, and $r = 1500 m$. The circumference of the storage ring is $2\pi r = 2\pi (1500 m) = 9424.8 m$. The speed of the muons is:$$v = \frac{2\pi r}{t} = \frac{2\pi (1500 m)}{2.2 	imes 10^{-6} s} = 0.9994c$$The Lorentz factor is:$$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{1}{\sqrt{1 - (0.9994c)^2/c^2}} = 22.36$$The time dilation factor is:$$t' = \frac{t}{\gamma} = \frac{2.2 	imes 10^{-6} s}{22.36} = 9.84 	imes 10^{-8} s$$The number of trips around the storage ring that the muons make before they decay is:$$n = \frac{t'}{t} = \frac{9.84 	imes 10^{-8} s}{2.2 	imes 10^{-6} s} = 0.0447$$Therefore, the correct answer is A.

Question 14

An Earth observer sees a spaceship at an altitude of 980 m moving downward toward the Earth at a speed of 0.600c.What is the spaceship's altitude as measured by an observer in the spaceship?

Accepted Answer

The altitude as measured by an observer in the spaceship is contracted due to length contraction. The formula for length contraction is L = L0 * sqrt(1 - v^2/c^2), where L0 is the proper length (980 m), v is the velocity (0.600c), and c is the speed of light. Calculating this gives approximately 784 m.

Question 15

A tuning fork has a frequency of 400 Hz and hence a period of 2.50 × 10−³ s.If the tuning fork is in an inertial frame of reference moving by the observer at speed of 0.680c,what is the frequency of the fork as measured by the observer? (Assume that measurements are strictly by optical means and that the speed of sound waves in air is not pertinent here. )

Accepted Answer

The answer of A tuning fork has a frequency of...

Question 16

A meterstick moving in a direction parallel to its length appears to be only 60.0 cm long to an observer.What is the meterstick's speed relative to the observer? (c = 3.00 × 10⁸ m/s)

Accepted Answer

We will use the length contraction formula:

L' = L * sqrt(1 - (v^2/c^2))

where L is the proper length and L' is the contracted length.

60.0 cm = 100 cm * sqrt(1 - (v^2/c^2))

Solving for v, we get:

v = sqrt((1 - (60.0 cm/100 cm)^2) * c^2)

v = 2.40 × 10^8 m/s

Therefore, the answer is Choice B.

Question 17

A mass is bouncing on the end of a spring with a period T when measured by a ground observer.What would the period of oscillation be (as measured by the same observer)if the mass and spring were moving past the ground observer at a speed of 0.600c?

Accepted Answer

When an observer is moving at a velocity comparable to the speed of light, the observed period of oscillation is longer due to time dilation. The formula for time dilation is:
t' = t / sqrt(1 - v^2/c^2)
where t is the time measured in the rest frame (ground observer) and v is the velocity of the moving frame (mass and spring). Plugging in the given values, we get:
t' = T / sqrt(1 - (0.600c)^2/c^2) = 1.25T
Therefore, the period of oscillation as measured by the moving observer is 1.25 times the period measured by the ground observer. The best choice is B.

Question 18

What is proper time?

Accepted Answer

Proper time is the time measured by a clock at rest with respect to the observer, as opposed to coordinate time, which is the time interval between two events measured in a reference frame in which they occur at the same location. This concept is important in special and general relativity.

Question 19

The period of a pendulum is 2.0 s in a stationary inertial frame of reference.What is its period when measured by an observer moving at a speed of 0.80c with respect to the inertial frame of reference?

Accepted Answer

Due to time dilation in special relativity, the period of the pendulum will appear longer to an observer moving at a high speed relative to the pendulum. The time dilation factor is given by the Lorentz factor, which is greater than 1 for speeds close to the speed of light.

Question 20

How fast would a rocket have to move past a ground observer if the latter were to observe a 3.0% length shrinkage in the rocket length? (c = 3.00 × 10⁸ m/s)

Accepted Answer

The length contraction formula is: 
L' = L√(1 - (v^2/c^2))
Where L is the original length, L' is the contracted length, v is the velocity of the rocket relative to the ground observer, and c is the speed of light.

Plugging in the given values, we get:
0.97L = L√(1 - (v^2/c^2))

Simplifying, we get:
0.03 = √(1 - (v^2/c^2))

Squaring both sides, we get:
0.0009 = 1 - (v^2/c^2)
v^2/c^2 = 0.9991

Taking the square root and multiplying by c, we get:
v = 0.73 × 10^8 m/s or 0.73c

Therefore, the answer is choice C.

Quiz 27: Quantum Physics Blackbody Radiation and Plancks Hypothesis: Part A

The astronaut whose heart rate on Earth is 60 beats/min increases his velocity to v = 0.60c.Now what is his heart rate as measured by an Earth observer?

If astronauts could travel at v = 0.90c,we on Earth would say it takes (4.2/0.90) = 4.7 years to reach Alpha Centauri,4.2 lightyears away.The astronauts disagree.How much time passes on the astronaut's clocks?

A muon formed high in the Earth's atmosphere travels at a speed 0.990c for a distance of 6.40 km before it decays.What is the muon's lifetime as measured in its reference frame?

A space probe has a 24.7-m length when measured at rest.What length does an observer at rest measure when the probe is going by at a speed of 0.860c?

A ground observer measures the period of a pendulum moving as a part of an inertial frame of reference to be 2.30 s as the inertial frame moves by at a velocity of 0.800c.What would the observed period be of the same pendulum if its inertial frame were at rest with respect to the observer?

The observed relativistic length of a super rocket moving by the observer at 0.57c will be what factor times that of the measured rocket length if it were at rest?

A spaceship is moving away from an observer at relativistic speed.Its length at rest is 100 m,but the observer measures its length as 67 m.A time interval of 100 s on the spaceship would be measured by the observer to be which of the following?

An astronaut is traveling at a high rate of speed from the Earth to a distant star system.Observers here on Earth note the relativistic effect that the astronaut's clocks run slow.What relativistic effect,if any,does the astronaut observe?

A rocket ship is 74.0 m in length when measured before leaving the launching pad.What would its velocity be if a ground observer measured its length as 60.0 m while it is in flight? (c = 3.00 × 10⁸ m/s)

A muon formed high in Earth's atmosphere travels at a speed 0.9900c for a distance (as we see it) of 9200 m before it decays.How far does the muon travel as measured in its frame?

An astronaut at rest has a heart rate of 65 beats/min.What will her heart rate be as measured by an earth observer when the astronaut's spaceship goes by the earth at a speed of 0.80c?

A knight on horseback holds a 10-m lance.The horse can run at 0.78c.(It wins most of its races!) How long will the lance appear to a person that is standing still on the ground as the horse runs past?

Muons at speed 0.9994c are sent round and round a circular storage ring of radius 1500 m.If a muon at rest decays into other particles after an average T = 2.2 × 10−⁶ s,how many trips around the storage ring do we expect the 0.9994c muons to make before they decay?

An Earth observer sees a spaceship at an altitude of 980 m moving downward toward the Earth at a speed of 0.600c.What is the spaceship's altitude as measured by an observer in the spaceship?

A meterstick moving in a direction parallel to its length appears to be only 60.0 cm long to an observer.What is the meterstick's speed relative to the observer? (c = 3.00 × 10⁸ m/s)

A mass is bouncing on the end of a spring with a period T when measured by a ground observer.What would the period of oscillation be (as measured by the same observer) if the mass and spring were moving past the ground observer at a speed of 0.600c?

What is proper time?

The period of a pendulum is 2.0 s in a stationary inertial frame of reference.What is its period when measured by an observer moving at a speed of 0.80c with respect to the inertial frame of reference?

How fast would a rocket have to move past a ground observer if the latter were to observe a 3.0% length shrinkage in the rocket length? (c = 3.00 × 10⁸ m/s)

Quiz 27: Quantum Physics Blackbody Radiation and Plancks Hypothesis: Part A

The astronaut whose heart rate on Earth is 60 beats/min increases his velocity to v = 0.60c.Now what is his heart rate as measured by an Earth observer?

If astronauts could travel at v = 0.90c,we on Earth would say it takes (4.2/0.90) = 4.7 years to reach Alpha Centauri,4.2 lightyears away.The astronauts disagree.How much time passes on the astronaut's clocks?

A muon formed high in the Earth's atmosphere travels at a speed 0.990c for a distance of 6.40 km before it decays.What is the muon's lifetime as measured in its reference frame?

A space probe has a 24.7-m length when measured at rest.What length does an observer at rest measure when the probe is going by at a speed of 0.860c?

A ground observer measures the period of a pendulum moving as a part of an inertial frame of reference to be 2.30 s as the inertial frame moves by at a velocity of 0.800c.What would the observed period be of the same pendulum if its inertial frame were at rest with respect to the observer?

The observed relativistic length of a super rocket moving by the observer at 0.57c will be what factor times that of the measured rocket length if it were at rest?

A spaceship is moving away from an observer at relativistic speed.Its length at rest is 100 m,but the observer measures its length as 67 m.A time interval of 100 s on the spaceship would be measured by the observer to be which of the following?

An astronaut is traveling at a high rate of speed from the Earth to a distant star system.Observers here on Earth note the relativistic effect that the astronaut's clocks run slow.What relativistic effect,if any,does the astronaut observe?

A rocket ship is 74.0 m in length when measured before leaving the launching pad.What would its velocity be if a ground observer measured its length as 60.0 m while it is in flight? (c = 3.00 × 108 m/s)

A muon formed high in Earth's atmosphere travels at a speed 0.9900c for a distance (as we see it) of 9200 m before it decays.How far does the muon travel as measured in its frame?

An astronaut at rest has a heart rate of 65 beats/min.What will her heart rate be as measured by an earth observer when the astronaut's spaceship goes by the earth at a speed of 0.80c?

A knight on horseback holds a 10-m lance.The horse can run at 0.78c.(It wins most of its races!) How long will the lance appear to a person that is standing still on the ground as the horse runs past?

Muons at speed 0.9994c are sent round and round a circular storage ring of radius 1500 m.If a muon at rest decays into other particles after an average T = 2.2 × 10−6 s,how many trips around the storage ring do we expect the 0.9994c muons to make before they decay?

An Earth observer sees a spaceship at an altitude of 980 m moving downward toward the Earth at a speed of 0.600c.What is the spaceship's altitude as measured by an observer in the spaceship?

A meterstick moving in a direction parallel to its length appears to be only 60.0 cm long to an observer.What is the meterstick's speed relative to the observer? (c = 3.00 × 108 m/s)

A mass is bouncing on the end of a spring with a period T when measured by a ground observer.What would the period of oscillation be (as measured by the same observer) if the mass and spring were moving past the ground observer at a speed of 0.600c?

What is proper time?

The period of a pendulum is 2.0 s in a stationary inertial frame of reference.What is its period when measured by an observer moving at a speed of 0.80c with respect to the inertial frame of reference?

How fast would a rocket have to move past a ground observer if the latter were to observe a 3.0% length shrinkage in the rocket length? (c = 3.00 × 108 m/s)

A rocket ship is 74.0 m in length when measured before leaving the launching pad.What would its velocity be if a ground observer measured its length as 60.0 m while it is in flight? (c = 3.00 × 10⁸ m/s)

Muons at speed 0.9994c are sent round and round a circular storage ring of radius 1500 m.If a muon at rest decays into other particles after an average T = 2.2 × 10−⁶ s,how many trips around the storage ring do we expect the 0.9994c muons to make before they decay?

A meterstick moving in a direction parallel to its length appears to be only 60.0 cm long to an observer.What is the meterstick's speed relative to the observer? (c = 3.00 × 10⁸ m/s)

How fast would a rocket have to move past a ground observer if the latter were to observe a 3.0% length shrinkage in the rocket length? (c = 3.00 × 10⁸ m/s)