Theoretical Mechanics: Fall-2021
Practice Zapping with d: Due Day 12

1. Shooting a Box Suppose the spaceship is approaching the Earth at \(3c/5\) and shoots a box at a speed of \(4c/5\) (from the spaceship's perspective).
   1. At what velocity will an Earth-bound observer see the box if it is shot directly towards the Earth?
   2. Sensemaking: Reasonable Value
      Briefly discuss why (or why not) your answer seem reasonable.
2. Zapping with "d" Zap the following equations with “d”:

   1. \(f= 3x^3-e^{2x}\)

   2. \(g = \sin(y^2)\)

   3. \(5z^4 = \ln(p)\)
3. Relativistic Explosion A lump of clay with mass \(M\) explodes into two equal pieces \(m\) moving directly away from each other.
   1. What is the speed of each of the pieces after the explosion?
   2. Sensemaking: Check a Special Case
      Conceptually, what would you expect the speed of the lumps to to be if \(m=\frac{M}{2}\)? Show that your answer to part (a) works for this special case.
   3. What is the relativisic kinetic energy of each of the pieces?
4. Moving Clocks Run Slow

   Imagine that you have a light clock with a period T (1 “tick” takes time \(T\)).

   Another observer moves relative to you with velocity \(+V\)

   1. Draw a spacetime diagram of 1 tick of the your light clock.
   2. What are the spacetime coordinates of the end of the “tick” of your light clock in:

      1. Your reference frame?

      2. The other observer's frame?

   3. Now image that the other observer has an identical light clock. Draw a new spacetime diagram of 1 “tick” of the other observer's light clock.
   4. What are the spacetime coordinates of the end of the “tick” of the other observer's light clock in:

      1. The other observer's frame?

      2. Your reference frame?

   5. Sensemaking: Tell a Conceptual Story
      Imagine that one of your classmates says, “Moving clocks run slow.” Do you agree? Explain your reasoning.