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Time Slows Down

Explore the concept of time dilation and the relationship between different frames of reference, using the example of a moving clock and the observed slowing down of time. Discover the formula for time dilation and its implications on the perception of time.

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Time Slows Down

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  1. This line marks simultaneous later times. Time Slows Down t’ t x’ The x axis marks all the places where t = 0. x

  2. This line marks equal or simultaneous times t’, in the other frame, F’. The x’ axis marks all the points where t’ = 0. t t’ x’ x

  3. Consider the point where the ‘stationary’ frame’s time hits the t’ axis. We can’t assume that t = t’. Let’s assume for now that t = gt’. t t’ x’ x

  4. Consider the point where the ‘moving’ frame’s time hits the t axis. We’ll label the two times as t1 and t2. All frames are equivalent, so t’ = g t1. t’ t2 t1 x’ x

  5. t2 = g t’ How are t1 and t2 related? t’ = g t1 t2 = g2t1 t’ t2 t1 x’ x

  6. t2 = g2t1 What does that tell you about g ? What does that tell you about time? gis greater than one. t t’ t” x’ Each observer sees the other ‘moving’ observer with their time ticking more slowly. x

  7. We can find the formula for g by considering a ‘light’ clock, which sends light up and down between mirrors once every time t.

  8. Suppose that this clock is on a rocket ship moving at a speed v to the right. We observe that the light travels farther in our frame. To keep c constant, it must also take more time, c = Dx/Dt. Therefore, our time interval is larger than the rocket’s. Their clock runs slower..

  9. = 1/ 1 – v2/c2 t = 1/ 1 – v2/c2 t’ c2 t2 = v2 t2 + c2 t’2 (c2 - v2 ) t2 = c2 t’2 How much more slowly can be found by applying Pythagoras’ theorem to the diagram below. Write the equation and solve for t. ct ct’ vt

  10. This formula does not depend on the fact that we used a ‘light’ clock. Anything that ‘ticks’ will do. If you were on a rocket ship and sent a signal to Earth every time your heart beat, doctors on Earth would say that your pulse was slow.

  11. If the Earth doctor sent you a signal every time that her heart beat, you would say that her pulse was a) faster b) normal c) slower

  12. Hint: All inertial frames are equivalent. a) faster b) normal c) slower

  13. = 1/ 1 – v2/c2 What is g, if the relative speed of the two frames is 3/5 c? t t’ t” x’ x

  14. What is t’, if t = 10? 10 • = 5/4 t t’ 8 x’ x

  15. 10 8 t’ is shorter but it ‘looks’ longer. t t’ x’ x

  16. Reality Check#3: Muons at CERN were accelerated to high speeds and lived 20 times longer than normal.

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