Conceptual Physics

Strategy

1. Use the Pythagorean theorem to determine the distance the light pulse moves in
   one tick as measured by Katherine.


2. Use the fact that distance equals the product of speed and time to replace the
   distances in the diagram.


3. Simplify the equation.

Mathematics principle

We will use the Pythagorean theorem

Step-by-step derivation

In Katherine’s reference frame, the light traces out the hypotenuses of two right

triangles. We use the Pythagorean theorem to calculate the hypotenuse. The triangle’s

height is calculated using the speed of light and the time measured by the professor. Its

base is calculated using Katherine’s measurement of the clock’s speed.

We have one last distance left, and again we substitute for it using the speed equation. Then a series of algebraic steps yield the equation for

time dilation.

Equation for time dilation

t = “stationary” observer time

t 0 = proper time in “moving” frame

v = speed of reference frame

c = speed of light

Step Reason

1. Pythagorean theorem

2. light travels two hypotenuses

3.^2 h = ct 0 definition of speed

4. 2 L = vt definition of speed

5. substitute equations 3 and 4 into

equation 2

Step Reason

6. 2 s = ct definition of speed

7. substitute equation 6 into equation 5; square both sides

8. expand and re-arrange

9. divide both sides by c^2 – v^2 ; simplify

10. take square root

(^646) Copyright 2007 Kinetic Books Co. Chapter 35