Question

1. Here is a slight variation of the train/tunnel problem. A very fast runner is carrying a pole that is 20m in length when at rest. The runner runs toward a barn that is 10m in length when at rest. We will call event A the time/location where the leading edge pole enters the barn. Event B is when the trailing edge of the pole enters the barn. Event C is when the leading edge of the pole leaves the barn. Event D is when the trailing edge of the pole leaves the barn.
   1. What is the velocity of the runner if an observer at the front door of the barn (and at rest with respect to the barn) “sees” the pole as 10m in length? (to three decimal places please)
   2. Consider the runner. What is the length of the barn as seen by the runner?
   3. Staying with the runner, Event A is 0m time and 0m location (0,0). Event B has coordinates (23.09,-20), (the time is length traveled divided by the velocity, in units of c, the location is the back of the pole). Using this same logic, find the coordinates of events C and D.
   4. Now consider the observer at the front of the barn. Event A is still (0,0). Event B is measured at (11.55,0). Note the interval is conserved (23.09² – 20² = 11.55²). Find the observers coordinate for events C and D.
   5. What is the temporal order of the events as seen by the runner, and by the observer?

Answer 1

a) The velocity of the runner with respect to the observer be , then using the formula for length contraction the length of the pole w.r.t. the observer will be

$L= L_{0}\sqrt{1-\frac{v^{2}}{c^{2}}} \quad (1)$

$L_{0} = 20\text{ m}$ is the rest length of the pole. Hence

$\frac{10}{20} = \sqrt{1-\frac{v^{2}}{c^{2}}}\Rightarrow \frac{v}{c} = \frac{\sqrt{3}}{2}\Rightarrow v = 0.866c \quad (2)$

b) In runner's frame of reference the barn is coming towards it with velocity
v= 0.866c
. Hence in runner's frame of reference, barn will be half of its rest length i.e. $5\text{ m}$ .

c) The event C is when the leading edge of the pole leaves the barn. The time taken by the back end of the barn to reach the leading edge of the pole is

$t_{C,runner} = \frac{5\text{ m}}{0.866} = 5.77\text{ m} \quad (3)$

Hence the coordinates of event C in runner's frame are . Event D is when the back end of the barn reach the trailing edge of the pole. To reach the trailing edge of the pole the back end of the barn has to cover a distance of

$d = \text{ length of the pole }+\text{ length of the barn } = 20\text{ m}+5\text{ m} = 25\text{ m} \quad (4)$

Hence the time taken by the back end of the barn to reach the trailing edge of the pole is

$t_{D,runner} = \frac{25\text{ m}}{0.866} = 28.87 \text{ m} \quad (5)$

The coordinates of event D in runner's frame are .

d) In barn's frame of reference the length of the barn is $10\text{ m}$ and the length of the pole is also $10\text{ m}$ . The time taken by the leading edge of the pole to reach the back end of the barn is

$t_{C,barn} = \frac{10\text{ m}}{0.866} = 11.55\text{ m} \quad (6)$

The coordinates of event C in barn's frame of reference are . To reach the back end of the barn, the trailing edge of the pole has to cover a distance of

$d = \text{ length of the pole }+\text{ length of the barn } = 10\text{ m}+10\text{ m} = 20\text{ m} \quad (7)$

Hence the time taken by the trailing edge of the pole to reach the back end of the barn is

$t_{D,barn} = \frac{20\text{ m}}{0.866} = 23.09 \text{ m} \quad (8)$

The coordinates of event D in barn's frame of reference are .

e) As seen by the runner the temporal order of events is:

First the leading edge of the pole enters the barn (event A). Then the leading edge of the pole leaves the barn (event C). Then the trailing edge of the pole enters the barn (event B), and finally the trailing edge of the pole leaves the barn (event D).

As seen by the observer the temporal order of events is:

First the leading edge of the pole enters the barn (event A). Then the leading edge of the pole leaves the barn (event C) and the trailing edge of the pole enters the barn (event B), simultaneously. Finally the trailing edge of the pole leaves the barn (event D).