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Lorentz transformation and muon experiment - class-XI

Description: lorentz transformation and muon experiment
Number of Questions: 20
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Tags: physics option a: relativity
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The minute hand of the clock is 4 cm long. The average velocity of the tip of the minute hand between 11:00 am to 11:30 am is:

lorentz transformation and muon experiment option a: relativity physics

  1. $1.5\times { 10 }^{ -5 }{ m }/{ s }$

  2. $4.4\times { 10 }^{ -5 }{ m }/{ s }$

  3. $1.8\times { 10 }^{ -6 }{ m }/{ s }$

  4. $3.5\times { 10 }^{ -6 }{ m }/{ s }$

Show answer
Correct Option: B
Explanation:

Average velocity $ = \dfrac{Total \; Displacement}{Total \; Time}$


$ = \dfrac{  2\times 0.04}{30 \times 60} = 4.44 \times 10^{-5} \;m/s$

Suppose a new planet is discovered, which revolves around the sun (i.e. it is a part of our solar system). If its lies between saturn and Jupitar, its time period of revolution would be 

lorentz transformation and muon experiment option a: relativity physics

  1. Less than that of Jupiter

  2. More than that of Jupiter

  3. Equal to that of Jupiter or saturn

  4. More than that of saturn.

Show answer
Correct Option: A
Explanation:

We don't know the exact order of all the planets, but Jupiter formed first, with Saturn close behind. Both were finished in less than 3 million years. The inner planets (Mercury, Venus, Earth, Mars) got off to a late start.

We don't observe space contraction in our every day lives and the reason is
lorentz transformation and muon experiment option a: relativity physics

  1. Space contraction is not noticeable until objects approach the speed of light.

  2. Space contraction does not occur until objects approach the speed of light.

  3. Space contraction only occurs in a vacuum.

  4. Space contraction is offset by the effect of gravity.

  5. Acceleration is necessary for length contraction.

Show answer
Correct Option: A
Explanation:

The contracted length of objects moving relative to an observer with speed $v$ is given by $l'=l _0\sqrt{1-(\dfrac{v}{c})^2}$.

Hence at small speeds $v<<c$, $l'\approx l _0$, and hence the space contraction is not noticeable.

The best definition for space contraction ?
lorentz transformation and muon experiment option a: relativity physics

  1. The length of an object decreases in the same direction it is moving.

  2. All dimensions of an object decrease at high speeds.

  3. The length of an object increases in the same direction it is moving.

  4. Both length and width, but not depth, decrease at high speeds.

  5. Space is constant regardless of its location or state of motion.

Show answer
Correct Option: A
Explanation:

When an object is moving relative to an observer, the length of the object as observed by a moving observer is less than that observed by a stationary one in the direction of the movement.

A rod of rest length L moves at a relativistic speed. Let L' = L/$\gamma$. Its length

lorentz transformation and muon experiment option a: relativity physics

  1. must be equal to L'

  2. may be equal to L

  3. may be more than L' but less than L

  4. may be more than L

Show answer
Correct Option: B,C

A man is sitting inside a moving train and observes the objects outside of the train. Then choose the single correct choice from the following statements

lorentz transformation and muon experiment option a: relativity physics

  1. all stationary objects outside the train will move with same velocity in opposite direction of the train with respect to the man.

  2. Stationary objects near the train will move with grater velocity & object far from train will move with lesser velocity with respect to the man

  3. large objects like moon or mountains will move with same velocity as that of the train

  4. all of these.

Show answer
Correct Option: A
Explanation:

All small objects will move in opposite direction of train with same velocity.

Far objects like moon will appear stationary.

According to special relativity, which of the following people would see me aging most slowly? (I am sitting)

lorentz transformation and muon experiment option a: relativity physics

  1. A person running by me at $6.0\ m/s$

  2. A person sitting next to me in a rocket travelling at half the speed of light

  3. A person viewing me from the top of a mountain

  4. A person beside me on a bus moving at $15 m/s$

  5. All people would see me age at the same rate

Show answer
Correct Option: B
Explanation:
A person sitting next to me in a rocket travelling at half the speed of light will see the person ageing more slowly, as faster an object is accelerated to, the higher the degree of time dilation effect relative to an object of lower speed.
Einstein's theory of relativity works on a human scale: the higher you are, the faster you age.

A person is watching a rocket with an astronaut inside move by at a speed near the speed of light.
Which of the following statement is true?

lorentz transformation and muon experiment option a: relativity physics

  1. The passage of time on the astronaut's watch is faster from the person's perspective than from the astronaut's

  2. The passage of time on the astronaut's watch is the same from the perspective of the person and the astronaut

  3. The passage of time on the astronaut's watch is faster from the perspective of the astronaut than from the perspective of the person

  4. The passage of time on the person's watch is faster, from his own perspective, as the rocket files by, than it was before the rocket flew by

  5. The passage of time on the astronaut's watch is faster, from his own perspective, as he flies by the person, than it was before he flew by the person

Show answer
Correct Option: A
Explanation:

Passage of time as measured by the person in rest frame           $t = \dfrac{\tau}{\sqrt{1-v^2/c^2}}$      $\implies t>\tau$

where  $\tau$ is the passage of time as measured by the astronaut in moving frame. 
$\implies$ Watch in a moving frame runs at a slower rate than that in rest frame.
Thus passage of time on the astronaut's watch is faster from the person's perspective than from the astronaut's perspective. 

The length of cruiser is $200 m$. If If the cruiser travels at a speed of $(\dfrac{\sqrt{3}}{2})c$ past a planet. Calculate the length of the cruiser measured by the inhabitants of the planet.

lorentz transformation and muon experiment option a: relativity physics

  1. $0$

  2. Between $0$ and $200 m$

  3. $200 m$

  4. Greater than $200 m$

  5. None of the above, since it is impossible to reach the described speed

Show answer
Correct Option: B
Explanation:

According to the theory of relativity, the length of objects observed by a moving frame is less as compared to that observed by the rest frame. This is called length contraction.

The planet for the cruiser is a moving frame, and the cruiser itself is rest frame for it. Thus the length of cruiser as observed by observer on planet is less than $200m$.

A space traveler is moving at a speed of 0.6 times the speed of light as seen from the Earth's frame of reference. After one year passes on Earth, the space traveler will
lorentz transformation and muon experiment option a: relativity physics

  1. age more than one year

  2. age less than one year

  3. age exactly one year

  4. not age at all

  5. become younger

Show answer
Correct Option: B
Explanation:

Given :   $v = 0.6 c$

The time elapsed on earth         $t  =1$ year.
Using relativity, time elapsed in the space traveler frame         $\tau = t \sqrt{1-\beta^2}$         where  $\beta = v/c  = 0.6$
$\therefore$   $\tau = 1 \times  \sqrt{1- (0.6)^2}  =1\times  0.8  = 0.8$ year                      $\implies$   $\tau < t$
Thus times elapsed in space traveler frame is less than that of earth, so space traveler will age less than one year.

An astronaut on a fast-moving spaceship appears to age only $1$ year to an outside observer, even though the person travels for $5$ years from the observer's perspective. The astronaut travels a distance of X during this time, from the observer's perspective.
Which of the following is true from the astronaut's perspective?

lorentz transformation and muon experiment option a: relativity physics

  1. The astronaut ages $5$ years during the trip

  2. The distance the astronaut travels is X

  3. The distance the astronaut travels is greater than X

  4. The distance the astronaut travels is less than X

  5. The astronaut ages more than $5$ years during the trip

Show answer
Correct Option: D
Explanation:
Time of journey from the person's perspective, $t = 5$ year
Thus distance traveled from the person's perspective, $X = vt =  5v$ 

Time of journey from the astronaut's perspective,  $t' = 1$ year
Thus distance traveled from the astronaut's perspective, $d' = vt' =  v$   $\implies$   $d'<X$ 

A person is watching a rocket with a astronaut inside move by at a speed near the speed of light. Which of the following statement is true?

lorentz transformation and muon experiment option a: relativity physics

  1. The length of the rocket is greater from the person's perspective than from the astronaut's perspective

  2. The length of the rocket is the same from the perspective of the person and the astronaut

  3. The length of the rocket is greater from the perspective of the astronaut than from the perspective of the person

  4. The person's length is greater, from his own perspective, as the rocket flies by, than it was before the rocket flew by

  5. The astronaut's length is greater, from his own perspective, as he flies by the person, than it was before he flew by the person

Show answer
Correct Option: C
Explanation:

Length of the moving rocket as seen by the person in rest frame       $L = L _o \sqrt{1-v^2/c^2}$     $\implies L<L _o$

where $L _o$ is the rest length of the rocket as seen by the astronaut in moving frame.
Thus length of the rocket is greater from the astronaut's perspective than from the person's perspective.

Two clocks are running in synchronism. One clock is moved at a very high speed and returned to the original position on the earth while the other remains at rest. The time passed in the travelling clock is 1 hr. Which of the following could be the time passed in the clock kept on the earth.
lorentz transformation and muon experiment option a: relativity physics

  1. 30 minutes

  2. 45 minutes

  3. 59 minutes

  4. 1 hours

  5. 2 hours

Show answer
Correct Option: E
Explanation:

According to the time dilation concept of relativity, the moving clock runs at slower rate than the rest clock  i.e rest clock measures longer time taken by an event than the moving clock.

Hence time elapsed in the clock kept on earth must be greater than 1 hr.
So, option E is correct.

An experimenter measures the length of a rod. In the cases listed, all motions are with respect to the lab and parallel to the length of the rod. In which of the cases the measured length will be minimum?

lorentz transformation and muon experiment option a: relativity physics

  1. The rod and the experimenter move with the same speed v in the same direction.

  2. The rod and the experimenter move with the same speed v in opposite directions.

  3. The rod moves at speed v but the experimenter stays at rest.

  4. The rod stays at rest but the experimenter moves with the speed v.

Show answer
Correct Option: B
Explanation:
The rod and the experimenter move with the same speed v in opposite directions, as if a rod is moving with speed parallel to its length, and when the rod and the experimenter move with the same speed $\nu$ in opposite direction the new $\nu$ will be $2\nu$, as the velocity adds up and hence the length will be minimum in this case.

If the speed of a rod moving at a relativistic speed parallel to its length is doubled,

lorentz transformation and muon experiment option a: relativity physics

  1. the length will become half of the original value

  2. the mass will become double of the original value

  3. the length will decrease

  4. the mass will increase

Show answer
Correct Option: C,D
Explanation:

Let the rest mass and rest length of the rod be $m _o$ and $L _o$, respectively.

According to relativity, length of the moving object parallel to its motion (or velocity) gets reduced by a factor of $\sqrt{1 - \dfrac{v^2}{c^2}}$ of its rest length but its mass gets increased by the same factor.
$\therefore$ Parallel length of the moving object         $L _{\parallel}  =L _o \sqrt{1-\dfrac{v^2}{c^2}}$         $\implies L _{\parallel} < L _o$
Also mass of the moving object         $m  =\dfrac{m _o}{ \sqrt{1-\dfrac{v^2}{c^2}}}$                $\implies m>m _o$
Hence options C and D are correct.

Imagine an unlikely situation where a cannon fires a cannon ball at $(0.7)c$ (seventy percent of the speed of light) relative to a train to which the cannon is attached. The train is moving at $(0.6)c$ (sixty percent of the speed of light) relative to the ground.
If an observer on the ground measured the speed of the cannon ball relative to the ground, what speed would he measure?

lorentz transformation and muon experiment option a: relativity physics

  1. Exactly $c$

  2. $(1.3)c$

  3. $(0.7)c$

  4. $(0.6)c$

  5. Between $(0.7)c$ and $c$

Show answer
Correct Option: E
Explanation:

Speed of train w.r.t ground       $v = 0.6c$

Speed of cannon ball w.r.t train      $v '  = 0.7 c$
According to relativity,  speed of cannon ball w.r.t ground       $v'' = \dfrac{v+v'}{1+vv'}$  
$\therefore$    $v'' = \dfrac{0.6c + 0.7c}{1 + (0.6c)(0.7c)} = \dfrac{1.3c}{1.42} = 0.915 c$

An experimenter measures the length of a rod. Initially the experimenter and the rod are at rest with respect to the lab. Consider the following statements.
(A) If the rod starts moving parallel to its length but the observer stays at rest, the measured length will be reduced.
(B) If the rod stays at rest but the observer starts moving parallel to the measured length of the rod, the length will be reduced.

lorentz transformation and muon experiment option a: relativity physics

  1. A is true but B is false.

  2. B is true but A is false.

  3. Both A and B are true.

  4. Both A and B are false.

Show answer
Correct Option: C

The length of the rod placed inside a rocket is measured as $1 m$ by an observer inside the rocket which is at rest. When the rocket moves with a speed of $36\times { 10 }^{ 6 }{ km }/{ hr }$ the length of the rod as measured by the same observer is :

lorentz transformation and muon experiment option a: relativity physics

  1. $0.997 m$

  2. $1.003 m$

  3. $1 m$

  4. $1.006 m$

Show answer
Correct Option: C
Explanation:

When the rocket moves with some velocity, the observer inside it moves with the same speed. 

Hence the relative velocity of the rod with respect to the observer remains zero. 
Hence he observes the length to be the rest length $=1\ m$

An air-bubble rises from the bottom of a long and narrow glass-tube full of glycerine. What happens to the speed of the air-bubble till it comes to the top?

lorentz transformation and muon experiment option a: relativity physics

  1. It goes on increasing

  2. It goes on decreasing

  3. It first increases, then moves up with constant velocity

  4. The speed remains the same throughout

Show answer
Correct Option: C

Box $1$ and box $2$ are identical when at rest relative to each other. An astronaut floating in intergalactic space sees the two boxes fly by from the astronaut's left to his right. Box $1$ flies by at $(0.8)c(80$% of the speed of light), and box $2$ flies by at $(0.9)c(90$% of the speed of light).
Because of relativistic effects, what is true from the point of view of the astronaut?

lorentz transformation and muon experiment option a: relativity physics

  1. Box $2$ is longer and taller than box $1$

  2. Box $2$ is longer and not as tall asbox $1$

  3. Box $1$ is longer and taller than box $2$

  4. Box $1$ is longer and not as tall as box $2$

  5. Box $2$ is shorter and the same height as box $1$

Show answer
Correct Option: E
Explanation:
According to relativity, the ;ength of moving object parallel to the direction of motion gets shortened but the length perpendicular to the motion remains the same.
Let the rest length (parallel to the motion) and rest height of both the boxes be $L _o$ and $h _o$ respectively.
Box 1 :     $v = 0.8c$
$\therefore$ Length of box 1 observed by the astronaut        $L' _1 = L _o \sqrt{1- v^2/c^2} = L _o \sqrt{1- (0.8)^2}  =0.6L _o$

Box 2 :     $v = 0.9c$
$\therefore$ Length of box 2 observed by the astronaut        $L' _2 = L _o \sqrt{1- v^2/c^2} = L _o \sqrt{1- (0.9)^2}  =0.44L _o$

$\implies$    $L _1' > L _2'$    and  $h' _1 = h' _2 = h _o$
Hence option E is correct.

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