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Consider a person inside a train, who is bouncing a ball up and down.

To him, the path of the ball looks straight up and down.

How do the path and the speed of the ball appear to an observer along the tracks outside the train? An observer outside the train would see the ball travel a longer path in one up-down cycle.

What would the path of the ball look like to an outside observer if the train was moving twice the speed in the previous example? The observer sees the ball going forward with the train at the same time that it is going up and down. So the ball’s path always slants forward. This also means that the observer measures faster speed for the ball ! The distance the ball travels in the same amount of time is longer for the observer. Because,

The longer the path of the ball, the faster the ball’s velocity.

If the train was moving even faster than in the previous example, would the ball appear to be moving faster or slower to a stationary observer? The faster the train is moving, the longer its path would be, Because the path of the ball would look even more slanted. Thus, a stationary observer would measure a faster velocity for the ball.

Lets apply this knowledge to relativity!

We can perform a thought experiment with a light beam replacing the ball.

Imagine that you and your friend Jackie are each in freely floating spaceships. You are moving at 70 % the speed of light. Hereinafter, 70% the speed of light will be written as 0.7c ; 90% the speed of light would be 0.9c . Because there are no nearby planets or other objects to use as a stationary reference, to you it looks like Jackie is moving at 0.7c while you are stationary. To Jackie, it looks like she is stationary and you are the one moving at 0.7c .

Jackie, then, shines a laser up to a mirror on her spaceship’s ceiling, and uses a clock to measure how long it takes the light to travel from the floor to the ceiling and back to the floor. Lets say that she has a microsecond clock and measures the light to travel up and down her spaceship in 10 microseconds (micro seconds = 10-6 seconds). How much time would it take for the light to bounce up and down in Jackie's ship according to you ? (i.e. what would you measure looking at Jackie's light?)

From your perspective, you are stationary and Jackie is moving at 0.7c. That means that you will see the light from Jackie’s laser travel on a slanted path (just like the previous example of the ball in a moving train). To you it will look like the light has traveled a longer path . Because the speed of light is constant and the distance that the light traveled is longer from your reference frame, this means that you would measure more time ! The figure shows that you would measure 15 microseconds compared to Jackie’s 10 microseconds. You see Jackie’s clock running at a slower rate than yours. This means that time must be passing more slowly on Jackie’s ship. This is called Time Dilation .

When you observe an object moving near light speed:

– Its time is slower relative to yours (you measure more time, because more time has passed).

You may also find the following explanation of Time Dilation (using Light Clocks) useful:

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Special_relativity_clocks_rods/index.html

How much slower is time running for Jackie? It depends on her speed relative to you. The faster she is moving, the more slanted the light path will appear to you and the greater the difference between the rate of her clock and yours.

Time runs more slowly in moving reference frames. The faster the other reference frame is moving, the slower the clocks tick within that reference frame.

We can calculate the exact amount of time that is seen in the moving frame by using the time dilation formula:

where tmoving is the time measured for the moving reference frame, trest is time measured in the reference frame of the observer, v is the velocity of the moving object, and c is the speed of light.

This can also be represented graphically and may help in visualizing how the formula works.

The moving frame is marked with a t’ in the Figure.

Lets do an example. Suppose Jackie is moving past you at 0.95c . How much time would pass for Jackie if 1 hour passes for you?

Before you plug in the values into the formula, think about what you expect to find. Should the time be running shorter or longer for Jackie? Because she is moving close to the speed of light, her clock would tick at a slower rate, meaning that she would measure less time.

We can convert this number to minutes by multiplying it by 60 min/hour and we get 18.6 minutes, so about 19 minutes. While an entire hour passes for you, Jackie measures only 19 minutes on her clock.

One thing to notice is to Jackie, everything looks normal and she doesn’t perceive her own time to be ticking slower. From Jackie’s perspective, she is stationary while you are the one who is moving and she would measure your time to tick slower than hers.

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