Your friend zips by on an ultrafast skateboard traveling at $\frac{3}{5}$ light speed. When 100 nanoseconds have gone by on your watch, how much time will you say has elapsed on your friend’s watch?

A few days earlier, you accompanied your friend to the skateboard shop and remember that the board she bought was 3 feet long. As she zips by, you measure the length of the skateboard in motion. What length do you find?

Not only is your friend’s skateboard ultrafast, it is also equipped with clocks attached to its front and rear. As above, your friend says these two clocks are 3 feet apart. She also claims that the two clocks are synchronized with one another. As she zips by, however, you note that they are not synchronized. From your perspective, what is the difference in the clock readings and which clock is ahead? (Use the usual approximation $c = 1~ {\rm foot}/$ $\mbox{nanosecond}$ and express your answer in nanoseconds.)

Each of the two clocks on your friend’s skateboard is equipped with a light that flashes when the clock strikes noon. How much time elapses on your watch between the two light flashes? (Unless otherwise stated, all such questions refer to the time difference “post processing”–that is, do not concern yourself with the issue of light travel time to your eyes, etc.)

Your friend says that the two flashes of light described in the previous problem occur on opposite ends of her skateboard, and so she claims that they occurred 3 feet apart. As she zips by, you measure the distance between the locations, from your perspective, where the two flashes of light occurred. What do you find?

Your friend is also carrying an ultrapowerful BB gun that fires BBs at $\frac{2}{3}$ light speed. She fires the gun, pointing it in the direction she’s moving. How fast do you say the BBs travel?

Your friend measures the energy of the bullet she fired and finds the values $E = 10$ (where the units used here for energy are (kg) (feet/nanosecond)$^2$ . From this data and the known speed of the bullet, determine the momentum of the bullet according to your friend (express your answer in units of (kg) (feet/nanosecond)).

Assume that you and your friend are both located at the origin of your respective frames and that the space-time origins of the two frames agree–namely, $(t = 0, x = 0)$ agrees with $(t’ = 0, x’ = 0)$. Now, according to your friend, one of the BBs she fires hits a target that from her perspective is stationary and located 100 feet away. Assume that your friend fired that particular BB just as you and she were passing. What are the space-time coordinates from your perspective of the bullet hitting the target? Express your answer in the form $(t, x)$ with units of time in ns and space in feet.

When 100 nanoseconds have elapsed on your friend’s watch, how much time will she say has elapsed on your watch?

When 100 nanoseconds have elapsed on your friend’s watch, she immediately stops, turns around, and heads back toward you at the same speed of her outbound journey, $\frac{3}{5}$ light speed. Just as she starts the return journey, how much time will she say has elapsed on your watch since the journey began?

Later, when she passes by you, how much time will have elapsed on each of your watches? Give your answer in the form (Time on Your Watch, Time on Your Friend’s Watch), both expressed in ns.

An Earth colony on the planet Krypton reports that an asteroid is heading directly toward them at $\frac{20}{101}$ $c$. Assume that Earth, Krypton and the asteroid’s trajectory all lie on a straight line (one that we can thus take to be the $x$ -axis) and that the asteroid is between Earth and Krypton. Assume that Earth and Krypton are at rest relative to one another and that they are a distance 1000 light-minutes apart. Scientists on Earth devise a plan to save the colony: They will launch their top-secret ultrafast nuclear missile toward the asteroid, and set the timer on the missile so that it will explode just as it reaches the asteroid, destroying it. The missile has speed $\frac{99}{101}$ $c$ and is launched when the asteroid is 605 light-minutes from the colony on the planet Krypton.

Determine how many minutes the scientists should set on the missile’s timer.

From the perspective of the inhabitants of Krypton, at what distance from them will the explosion take place?

About a year later, Krypton reports exactly the same situation–an asteroid is heading directly toward them at $\frac{20}{101}$ $c$. Based on the first experience, though, scientists realize that they should address the situation a little differently: A more effective approach for protecting the colony is to have the missile detonate when, from the missile’s perspective, it is 79.79 light minutes beyond the asteroid (this helps to reduce the asteroid fragments that land on the colony). Assuming that in this new situation, the missile is again launched when the asteroid is 605 light-minutes from the colony, determine how many minutes the scientists should now set on the missile’s timer.