Relative velocity train problems

Question

A train starting from rest from a platform passes a pole the front part of the train crosses the pole with a velocity u and the real part of the train cross the pole with a velocity v. What is the velocity with which the middle portion of train will cross

Answer 1

Assuming constant acceleration of the train (probably true, but, yes, it should be stated somewhere), use basic kinematics equations for constant a:

 

v^2 - u^2 = 2ad, where d is the length of the train in this case. (Because when the train has gone a distance d = to its own length, then the rear part of the train reaches the pole. Solve for a:

 

a = [(v^2 - u^2)/2d]

 

and plug that into the same kinematics equation with distance now equal to d/2, because that's how far the train had traveled when its "middle portion" crosses the pole (call the velocity when the middle is at the pole "w"):

 

w^2 - u^2 = 2[(v^2 - u^2)/2d](d/2)

 

= [(v^2 - u^2)/2d]d = (v^2 - u^2)/2

 

Solve that for w and that should be the velocity needed:

 

w^2 - u^2 = (v^2 - u^2)/2 = (v^2)/2 - (u^2)/2

 

w^2 = (v^2)/2 + (u^2)/2, square root both sides

 

w = [(v^2)/2 + (u^2)/2]^(1/2)