Week 6, clip 5

Timepoint:  12:40.5

1) A ruler of length L' that is moving (blue) appears to a stationary (black) observer to have a length L = L' / gamma, where gamma = 1 / (1-v^2/c^2)^0.5.  Since v < c, we must have gamma > 1.  Therefore, the moving ruler appears shorter to the stationary observer than to the observer moving with the ruler.

What speed must the ruler be moving for the stationary observer to measure it at half its original length?  That is, what is v so that L = L' / 2?  Give your answer as a fraction of the speed of light, rounded to two significant digits.

2) Imagine a traveler in a spaceship moving at 0.1c (one-tenth the speed of light) for one year.  How much slower do external clocks (say, on Earth) appear to run to the traveler compared to the clock on her spaceship?  Give your answer in terms of a fraction of a year, rounded to three significant digits.

Answers:
1) The accepted range is [0.85,0.88].

Explanation:  
For L = L' / 2, and thus gamma = 2.

2 = 1 / (1-v^2/c^2)^0.5

(1-v^2/c^2)^0.5 = 0.5

v/c = (0.75)^0.5

v = 0.866c

2) The accepted range is [0.994,0.996]

Explanation:
T' = T / gamma, where gamma = 1 / (1-v^2/c^2)^0.5

For v = 0.1 c, we have 
gamma = 1 / (1-(0.1)^2)^0.5

gamma = 1 / (0.99)^0.5

gamma = 1 / (0.994987)

gamma = 1.0050378

T' = 0.994987 T

So if T = 1 year, then T' = 0.995 years, rounded to three significant digits.