A revised look at the bomb in the attache case

This project must be done by individuals.

Back in Homework Set 2, you may recall a situation in which two attorneys have lunch at a Chinese restaurant. Unknown to them, a time bomb has been secretly hidden in the attache case carried by one of them. In the homework problem, you were asked to figure out where and when the bomb would explode, and to draw a space-time diagram showing the events.

Well, that problem was rather simplified: it didn't mention the time dilation that would occur as the two lawyers walked or ran away from the restaurant after lunch. This project asks you do re-analyze the situation, this time taking time dilation into account.

Here's the scenario:

Bob and Larry, relativistic attorneys-at-law, have lunch at their favorite Chinese restaurant. They finish, the waiter gives them fortune cookies, and the two leave leave. All three people -- Bob, Larry, and Waiter -- agree that the lawyers leave the restaurant at time -11 meters and at location -13 meters.

Event 1: Bob and Larry leave the restaurant. This takes place at (t₁ = -11 m, x₁ = -13 m) according to all three observers.

So, we have three observers: the Waiter, who stays motionless at the restaurant; Bob, who walks to the right at v_Bob = 0.2 c; and Larry, who carries an attache as he runs to the right at v_Larry = 0.75 c. That means we'll have three frames of reference to consider. Part of your task will be to draw a space-time diagram, showing a variety of items and actors in the reference frame of the Waiter.

Make a space-time diagram which covers time values from -15 m to +30 m and space values from -15 m to +15 m. Label the axes.
Place a circle at the location of Event 1
Draw the world line of Bob as he walks to the right. Label it "Bob".
Write an equation for Bob's world line
Draw the world line of Larry as he runs to the right. Label it "Larry".
Write an equation for Larry's world line

As he walks away from the restaurant to the right at a speed of v_Bob = 0.2 c, Bob reads his fortune cookie. It says, "We placed a bomb in your partner's attache case. It will explode when his watch reads t = 24 meters."

"Oh, no!" cries Bob. He grabs his walkie-talkie and, as his watch reads t = -1 meter, screams into it, "Larry, drop the case and run!"

Event 2: Bob sends a radio message to Larry when Bob's watch reads t_Bob,2 = -1 m.

What is the time interval between events 1 and 2 measured by Bob?
Using time dilation, compute the time interval between events 1 and 2 as measured by the Waiter.
What is the time of event 2, as measured by the Waiter?
What is the location of event 2, as measured by the Waiter?
Draw a square at the location of event 2 on the diagram.

A radio wave shoots towards the right from Event 2. It flies to the right at the speed of light, eventually reaching Larry.

Draw a dashed line on the diagram, showing the radio wave. Extend it until it meets Larry's world line.
Place a triangle on the diagram at the location of event 3.
Write an equation for the path of the radio wave.

When the radio wave reaches Larry, he hears Bob's warning message. If it reaches him before the bomb explodes, he can drop the attache case and run for safety.

Event 3: The radio wave reaches Larry.

The space-time diagram shows events in the frame of the Waiter. Look at your diagram and estimate the time of event 3, according to the Waiter.
Use the equation for Larry's world line, and the equation for the radio wave's path, to compute the EXACT time of event 3, according to the Waiter.
What is the time interval between events 1 and 3, according to the Waiter?
Using time dilation, what is the time interval between events 1 and 3, according to Larry?
What does Larry's watch read when the radio wave reaches him?
Does the radio wave reach Larry before the bomb explodes?