IB Physics HL A.5 Galilean & Special Relativity

[Maximum number: 13]

This question is about time dilation and relativistic mass.

(a)

Two space stations X and Y are at rest relative to each other. The separation of X and Y as measured in their frame of reference is $1.80 \times 10^{11} \mathrm{~m}$ .

State what is meant by a frame of reference.

[ 1 ]

(b)

A radio signal is sent to both space stations in (a) from a point midway between them. On receipt of the signal a clock in X and a clock in Y are each set to read zero. A spaceship S travels between X and Y at a speed of 0.750 c as measured by X and Y . In the frame of reference of S , station X passes S at the instant that X 's clock is set to zero. A clock in S is also set to zero at this instant.

[ 9 ]

(i)

Calculate the time interval, as measured by the clock in X , that it takes S to travel from X to Y .

[ 2 ]

$\frac{1.80\times10^{11}}{0.750\times3\times10^{8}}=800\ \mathrm{s}$

(ii)

Calculate the time interval, as measured by the clock in S , that it takes S to travel from X to Y .

[ 2 ]

$\gamma=\frac{1}{\sqrt{1-0.750^{2}}}=1.51$

time $=\frac{800}{1.51}=530\ \mathrm{s}$

(iii)

Explain whether the clock in X or the clock in S measures the proper time.

[ 2 ]

(iv)

Explain why, according to S , the setting of the clock in X and the setting of the clock in Y does not occur simultaneously.

[ 3 ]

(c)

Muons are produced in the upper atmosphere of Earth and travel towards the surface of Earth where they are detected. Explain how, with reference to the situation described in (b), the production and detection of muons provide evidence for time dilation.

[ 3 ]

[Maximum number: 7]

This question is about the Michelson-Morley experiment.

The diagram shows the basic features of the Michelson-Morley experiment.

(a)

During the experiment, Michelson and Morley slowly rotated the apparatus through $90^{\circ}$ . Discuss, with reference to Galilean transformations, the changes that Michelson and Morley expected to observe during the experiment.

[ 4 ]

(according to a Galilean transformation) speed of light depends on the direction of motion (through the ether);
beam which reflects from mirror $M_{1}$ takes different/less time than the beam reflecting from $\mathrm{M}_{2}$ ;
the time taken to travel each path will change as the apparatus is rotated;
a changing interference pattern is observed;

(b)

Describe how the observations made by Michelson and Morley differed from what was expected.

[ 1 ]

(c)

Explain the importance of the outcome of the Michelson-Morley experiment in support of the special theory of relativity.

[ 2 ]

[Maximum number: 10]

Two rockets, A and B , are moving towards each other on the same path. From the frame of reference of the Earth, an observer measures the speed of A to be 0.6 c and the speed of B to be 0.4 c. According to the observer on Earth, the distance between A and B is $6.0 \times 10^{8} \mathrm{~m}$ .

(a)

Define frame of reference.

[ 1 ]

(b)

Calculate, according to the observer on Earth, the time taken for A and B to meet.

[ 2 ]

(c)

Identify the terms in the formula.

u^{\prime}=\frac{u-v}{1-\frac{u v}{c^{2}}}

[ 1 ]

u and v are velocities with respect to the same frame of reference/Earth AND $u^{\prime}$ the relative velocity

Marking guidance:

Accept 0.4c and 0.6c for u and v

(d)

Determine, according to an observer in A , the velocity of B .

[ 2 ]

(e)

Determine, according to an observer in A , the time taken for B to meet A .

[ 2 ]

$\gamma=1.25$

so the time is t=1.6 «s»

(f)

Deduce, without further calculation, how the time taken for A to meet B , according to an observer in B , compares with the time taken for the same event according to an observer in A .

[ 2 ]

[Maximum number: 1]

This question is about time dilation and relativistic mass.

Two space stations X and Y are at rest relative to each other. The separation of X and Y as measured in their frame of reference is $1.80 \times 10^{11} \mathrm{~m}$ .

State what is meant by a frame of reference.

[Maximum number: 4]

This question is about the Michelson-Morley experiment.

The diagram shows the basic features of the Michelson-Morley experiment.

During the experiment, Michelson and Morley slowly rotated the apparatus through $90^{\circ}$ . Discuss, with reference to Galilean transformations, the changes that Michelson and Morley expected to observe during the experiment.

(according to a Galilean transformation) speed of light depends on the direction of motion (through the ether);
beam which reflects from mirror $M_{1}$ takes different/less time than the beam reflecting from $\mathrm{M}_{2}$ ;
the time taken to travel each path will change as the apparatus is rotated;
a changing interference pattern is observed;

[Maximum number: 2]

Two rockets, A and B , are moving towards each other on the same path. From the frame of reference of the Earth, an observer measures the speed of A to be 0.6 c and the speed of B to be 0.4 c. According to the observer on Earth, the distance between A and B is $6.0 \times 10^{8} \mathrm{~m}$ .

Calculate, according to the observer on Earth, the time taken for A and B to meet.

[Maximum number: 1]

An observer P sitting in a train moving at a speed v measures that his journey takes a time $\Delta t_{\mathrm{p}}$ . An observer Q at rest with respect to the ground measures that the journey takes a time $\Delta t_{\mathrm{Q}}$ .

According to Q there is an instant at which the train is completely within the tunnel. At that instant two lights at the front and the back of the train are turned on simultaneously according to Q .

The spacetime diagram according to observer Q shows event B (back light turns on) and event F (front light turns on).

Apply a Lorentz transformation to show that the time difference between events B and F according to observer P is $2.5 \times 10^{-7} \mathrm{~s}$ .

$\Delta t^{\prime}=1.25\times\frac{0.6\times100}{3\times10^{8}}=2.5\times10^{-7}\ \mathrm s$

[Maximum number: 2]

An observer P sitting in a train moving at a speed v measures that his journey takes a time $\Delta t_{\mathrm{p}}$ . An observer Q at rest with respect to the ground measures that the journey takes a time $\Delta t_{\mathrm{Q}}$ .

A second train is moving at a velocity of -0.70 c with respect to the ground.

Calculate the speed of the second train relative to observer P .

$u^{\prime}=\frac{-0.7-0.6}{1+0.7\times0.6}\,c=-0.92\,c$

[Maximum number: 3]

The space-time diagram shows the coordinate axes (x, c t) of an inertial reference frame S. Two events, P and Q, are plotted.

(a)

Calculate the space-time interval $(\Delta s)^{2}$ between P and Q .

[ 2 ]

Correct readoffs $\Delta x=3 \mathrm{~m}, c \Delta t=1 \mathrm{~m}$ $(\Delta s)^{2}=1^{2}-3^{2}=-8 \mathrm{~m}^{2}$

Marking guidance:

Ignore units as they are not required for the answer.

Do not award MP2 if the answer is positive

(b)

$(\Delta s)^{2}$ is an invariant quantity. Outline what is meant by an invariant quantity.

[ 1 ]

[Maximum number: 1]

An observer P sitting in a train moving at a speed v measures that his journey takes a time $\Delta t_{\mathrm{p}}$ . An observer Q at rest with respect to the ground measures that the journey takes a time $\Delta t_{\mathrm{Q}}$ .

State which of the two time intervals is a proper time.

$\Delta t_{\mathrm P}$ : observer sitting in the train

[Maximum number: 2]

An observer P sitting in a train moving at a speed v measures that his journey takes a time $\Delta t_{\mathrm{p}}$ . An observer Q at rest with respect to the ground measures that the journey takes a time $\Delta t_{\mathrm{Q}}$ .

Calculate the speed v of the train for the ratio $\frac{\Delta t_{P}}{\Delta t_{Q}}=0.30$ .

$\gamma=\frac{\Delta t_Q}{\Delta t_P}=\frac{1}{0.30}=3.3$ v=0.95 c

[Maximum number: 2]

An observer P sitting in a train moving at a speed v measures that his journey takes a time $\Delta t_{\mathrm{p}}$ . An observer Q at rest with respect to the ground measures that the journey takes a time $\Delta t_{\mathrm{Q}}$ .

Later the train is travelling at a speed of 0.60 c . Observer P measures the length of the train to be 125 m . The train enters a tunnel of length 100 m according to observer Q .

Show that the length of the train according to observer Q is 100 m .

$\gamma=1.25$

length of train according to Q $=125/1.25=100\ \mathrm m$

[Maximum number: 3]

An observer P sitting in a train moving at a speed v measures that his journey takes a time $\Delta t_{\mathrm{p}}$ . An observer Q at rest with respect to the ground measures that the journey takes a time $\Delta t_{\mathrm{Q}}$ .

According to Q there is an instant at which the train is completely within the tunnel. At that instant two lights at the front and the back of the train are turned on simultaneously according to Q .

The spacetime diagram according to observer Q shows event B (back light turns on) and event F (front light turns on).

Deduce, using the spacetime diagram, which light was turned on first according to observer P.

lines parallel to the $x^{\prime}$ axis and passing through B and F;
intersections on the $ct^{\prime}$ axis at $\mathrm B^{\prime}$ and $\mathrm F^{\prime}$ ;
light at the front of the train must have been turned on first

[Maximum number: 1]

An observer P sitting in a train moving at a speed v measures that his journey takes a time $\Delta t_{\mathrm{p}}$ . An observer Q at rest with respect to the ground measures that the journey takes a time $\Delta t_{\mathrm{Q}}$ .

According to Q there is an instant at which the train is completely within the tunnel. At that instant two lights at the front and the back of the train are turned on simultaneously according to Q .

The spacetime diagram according to observer Q shows event B (back light turns on) and event F (front light turns on).

Draw the time $c t^{\prime}$ and space $x^{\prime}$ axes for observer P's reference frame on the spacetime diagram.

axes drawn with correct gradients of $\frac{5}{3}$ for $ct^{\prime}$ and 0.6 for $x^{\prime}$

[Maximum number: 1]

The space-time diagram shows the coordinate axes (x, c t) of an inertial reference frame S. Two events, P and Q, are plotted.

Events P and Q occur simultaneously in an inertial reference frame $S^{\prime}$ . The clocks of S and $S^{\prime}$ are synchronized at $x=x^{\prime}=0$ .

Determine the speed of $S^{\prime}$ relative to S.

$\left\langle\frac{v}{c}=\frac{1}{3}\right.$ so $\| v=0.33 \mathrm{c}$ or $1.0 \times 10^{8} \mathrm{~m} \mathrm{~s}^{-1}$

Watch for ECF from bi.

Marking guidance:

Allow fraction as the answer, i.e. $\frac{c}{3}$

1

[Maximum number: 3]

This question is about time dilation and relativistic mass.

Muons are produced in the upper atmosphere of Earth and travel towards the surface of Earth where they are detected. Explain how, with reference to the situation described in (b), the production and detection of muons provide evidence for time dilation.

IB Physics HL A 5 Galilean and Special Relativity Questions

Exam points

Covered syllabus points

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (i)

Question (ii)

Question (iii)

Question (iv)

Question (c)

Question 2

Question (a)

Question (b)

Question (c)

Question 3

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)

Question (f)

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question (a)

Question (b)

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16