A train of proper length L moves with speed v with respect to the ground. When t

Question

A train of proper length L moves with speed v with respect to the ground. When the front of the train passes a tree on the ground, a ball is simultaneously (as measured in the ground frame) thrown from the back of the train toward the front, with speed u with respect to the train. What should u be so that the ball hits the front simultaneously (as measured in the train frame) with the tree passing the back of the train? Show that in order for a solution for u to exist, we must have v/c < (?5 -1)/2, which happens to be the inverse of the golden ratio.

Explanation / Answer

time taken by train to cross tree = v/L

speed of ball thrown raltive to ground = v+u

and distance covered by ball is L

so (v+u)*v/L = L

or u = L^2/v - v

and equation becomes

v^2 - u*v - L^2 = 0

for a solution to exist

we know b^2 - 4*a*c > 0

(-u)^2 - 4*1*L^2 >0

u^2 > 4*L^2

=== > u > 2L

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