PHYS 3310 Module 1.4

Length Contraction

Recommended Reading

1.4 Length Contraction

Learning Objectives

  • Explain how simultaneity and length contraction are related.
  • Describe the relation between length contraction and time dilation and use it to derive the length-contraction equation.

Practice!

Practice 1.4.1
What is the relationship between the proper length of an object, Lo, and the length measured by an observer moving relative to the object, L?
A. L0 > L
B. L0 = L
C. L0 < L
Check your answer: A
Practice 1.4.2
The spaceships of Bob and Alice sit side-by-side at a spaceport. They measure them and both agree that each spaceship is 26 meters long. Alice takes off and Bob observes Alice’s spaceship go by at a constant velocity such that the Lorentz factor is 2. While going by, Alice measures the length of her ship again. What value does she get?
A. 13 m
B. 26 m
C. 52 m
Check your answer: B
Practice 1.4.3
Same situation as in 1.4.2, but this time Bob measures the length of Alice’s ship as it flies by. What value does he get?
A. 13 m
B. 26 m
C. 52 m
Check your answer: A
Practice 1.4.4
Same situation as in 1.4.2, but this time Bob measures the length of his ship as Alice flies by. What value does he get?
A. 13 m
B. 26 m
C. 52 m
Check your answer: B
Practice 1.4.5
Same situation as in 1.4.2, but this time Alice measures the length of Bob’s ship as she flies by. What value does she get?
A. 13 m
B. 26 m
C. 52 m
Check your answer: A

Proper length L0 is the distance between two points measured by an observer who is at rest relative to both of the points.

Discuss!

Suppose you measure the length of a spaceship, at rest relative to you, to be 400 m.

(a) How long will you measure it to be if it flies past you at a speed of u = 0.75c?

(b) The spaceship has a large clock attached to its side. This clock ran at the same rate as your watch when you were in the same reference frame. How much time will pass on your watch as 80 s passes on the clock attached to the ship?

Check your answers: (a) 265 m (b) 121 s

Two spaceships named A and B are flying toward each other with relative speed of 0.800c.

(a) If the captain of ship A fires a missile, counts 10 s on his watch, and then fires a second missile, how much time will the captain of ship B measure to have passed between the firing of the two missiles?

(b) The captain of ship B knows that ship A uses 2-m-long missiles. She measures the length of the first missile, once it has finished accelerating, and finds it to be only 0.872 m long. What is the speed of the missile relative to ship B?

Check your answers: (a) 16.7 s (b) 0.900c