Workshop - Time dilation for objects in uniform motion

General directions

  1. Divide into working groups of three.
  2. Take each section and don't look ahead.
  3. Try to answer all of the questions and problems in each section.
  4. Observe the specified time limits.
  5. Be prepared to go over your results at the end of each section.

Exercise 1 - Light Clock (10 minutes)

Figure 1 above illustrates a light clock constructed from two opposing mirrors distance D apart. One "tick" of the light clock is the time it takes light to travel the distance D from one mirror to the other. As the light bounces back and forth between the mirrors we can imagine a ticking clock keeping track of our time.

  1. Suppose D is one meter. How many light clock ticks make one second?
  2. Give the formula for the time t shown by the light clock in terms of the speed of light c and the distance D.

Exercise 2 - Light Clock in Uniform Motion (10 minutes)

Figure 2 above is the view of the light path from an observer "on" the moving light clock. The light takes time t to complete its tick according to the observer on the light clock.

Figure 3 above adds to Figure 2 the view of the light path from an observer at rest relative to the moving light clock. The light takes time t' to complete its tick according to the observer on the embankment.

  1. Give the expression for the time t shown by the light clock for an observer on the light clock in terms of D and c.
  2. Let D' = c * t' in Figure 3. Give the expression for the time t' shown by the light clock from the view of the observer on the embankment in terms of D' and c.
  3. Compare the distance D = c * t traveled by the light as seen from the observer on the clock with the distance D' = c * t' as seen from the observer on the embankment. Which distance is greater? What does this say about the relative sizes of the time quantities t and t' representing the clock ticks from different points of view? (Hint: just look at the equations for D and D' and notice that c is the same in both)
  4. What is happening to time on the moving clock as observed from the embankment (compared to the time as seen on the moving clock) ?

Exercise 3 - Deriving the Time Dilation Factor (20 minutes)

The previous exercise showed that the time registered by a moving clock slowed down (the light clock ticks got bigger) when observed from the embankment. We want to determine how much the time slows down as the velocity of the moving clock increases. The time slow-down factor is called the time dilation factor. We will derive the time dilation factor from the diagram in Figure 3.
  1. Using Figure 3 and the Pythagorean Theorem, express a relationship between the distance D = c * t of light travel on the clock, the distance D' = c * t' of light travel as seen from the embankment, and the distance D'' = v * t' of clock travel during the clock tick t'.
  2. If you haven't already, express your Pythagorean relationship entirely in terms of c, v, t, and t' (no Ds).
  3. Now solve your equation for the clock tick time t' in terms of v, c and t. You will have to recall some of algebraic skills for manipulating equations. Your result should have the form t' = DT * t for some formula DT (the time dilation factor) expressed in terms of v and c. (Hint: first express t' squared in terms of c squared, t squared, and v squared).

Exercise 4 - Implications of the time dilation equation (15 minutes).

You have just derived the time dilation equation that specifies just exactly how much time slows down on moving objects relative to a stationary observer. The time dilation factor is the formula DT that translates the size of the time tick t as seen by the observer on the moving clock into the size of the time tick t' as seen by the observer on the embankment. Now consider the time dilation formula DT.

  1. What happens to the time dilation factor when v is small compared to c. (Hint: a very small fraction can be thought of as zero).
  2. What happens when v approaches c in the time dilation factor? (Hint: a fraction that has a very, very small denominator (bottom) compared to the numerator (top) approaches a very, very large number.
  3. What does this say about very, very fast moving objects?