When traveling past an observer with a relative speed v a ro

When traveling past an observer with a relative speed v, a rocket is measured to be 8.90 m long. When the rocket moves with a relative speed 2v, its length is measured to be 4.90 m .

a)What is the speed v? (in terms of c)

b) What is the proper length of the rocket?

Solution

A.

relative length contraction

L = L0*sqrt(1 - v^2/c^2)

for speed v, L = 8.9 m

for speed 2v, L = 4.9 m

using above equation

8.9*L0 = L0*sqrt(1 - v^2/c^2)

4.9*L0 = L0*sqrt(1 - 4v^2/c^2)

dividing bothequation

8.9/4.9 = sqrt(1 - v^2/c^2)/sqrt(1 - 4v^2/c^2)

3.299*(1 - 4v^2/c^2) = 1 - v^2/c^2

2.299 = 3.299*4*v^2/c^2 - v^2/c^2

2.299 = 12.196*v^2/c^2

v = c*sqrt(2.299/12.196) = 0.434*c

v = 0.434*c

B. L = L0*sqrt(1 - v^/c^2)

L0 = L/sqrt(1 - v^2/c^2)

L0 = 8.9/(sqrt(1 - 0.434^2)) = 9.878 m

proper length = 9.878 m