If 2 cars are traveling at the speed of light and the one in the back turns on its headlights, would the car in front be able to see them? Why or why not?
There is a problem with the question as stated. It is not possible for objects with mass to travel the speed of light. So, technically the question has no answer. But, if we were to restate the question slightly we would obtain some interesting results. If 2 cars are traveling nearly the speed of light and continue to get closer and closer to the speed of light, what happens when the car in the back turns on its headlights?
Two cases need to be considered. The first is what happens in the reference frame of the cars? Since the cars are not moving relative to each other, they could be treated as not moving at all. Then the light always takes the same time to catch the car in front. For example, if they are 300,000 km apart then the light will always take one second to catch up.
The more interesting case is when one considers a frame at rest relative to the cars. In this case, two things happen. First, the rest observer measures the distance between the cars to be shrinking the faster they go. Second, the rest observer measures the time it takes the light to go from the rear car to the front car to take longer the faster they go.
To find out how much we use the Lorentz transformations, Lr = Lm / g and Tr = gTm. Here Lr , Lm , Tr and Tm are rest frame length, moving frame length, rest frame time and moving frame time respectively. g equals 1 / ( 1 - v2/c2)˝ where v is the speed of the moving frame and c is the velocity of light. For example if they are going 99% the speed of light, the observer measures them to be 43,320 km apart, but the light takes 7.09 sec to travel the distance.
So, in the limit, the cars would be infinitesimally far apart and the light would take forever to get from the back car to the front car.
Submitted by J.(age 29, Texas, USA)
(September 29, 1997)