Proper time and formulas without numbers?

[sydboydell31]

Proper time and formulas without numbers?

(a) Write down the definition of proper time and explain how to use the appropriate formula with an example.

(b) Write down the definition of proper length and explain how to use the appropriate formula with an example.

(c) At what relative speed does a clock move if it runs at a rate that is one-half the rate of a clock at rest?

(d) At what relative speed does a meter stick move if its length is observed to shrink to 0.25 m?

(e) Two spaceships approach each other, each moving with the same speed as measured by a stationary observer on the Earth. Their relative speed is 0.70c. Determine the velocities of each spaceship as measured by the stationary observer on Earth.

 

[jbsteele]

A. Proper Time: Proper time is a measure of the elapsed time between two events as measured by a clock that moves along the same path as the observer. It is calculated by the formula t = ∫t_0^t dτ, where t_0 is the initial time and dτ is the proper time interval. For example, if a person is walking at a constant speed of 10 m/s for 10 seconds, the proper time interval would be dτ = 10 s/10 m/s = 1 s and the proper time would be t = ∫t_0^t dτ = 10 s. B. Proper Length: Proper length is a measure of the distance between two points along the same path as the observer. It is calculated by the formula L = ∫L_0^L dℓ, where L_0 is the initial length and dℓ is the proper length interval. For example, if the person from the previous example is walking for 10 meters, the proper length interval would be dℓ = 10 m/10 m/s = 1 s and the proper length would be L = ∫L_0^L dℓ = 10 m.C. The relative speed of the clock is 0.5c, where c is the speed of light.D. The relative speed of the meter stick is 0.75c.E. The velocities of each spaceship as measured by the stationary observer on Earth would be 0.35c.