Question

A train is moving with uniform acceleration. the two ends of the train pass through a point on the track with velocity v1 and v2. with what velocity the middle point of the train would pass through the same point

Answer 1

Let the train starts from A we with uniform velocity , at t₁ time train crosses the point B with speed v₁ and at t₂ time trains crosses the second point C with speed v₂ . Let D is the middle point between B and C .

A_________B-----------------D---------------C

Let the distance between B and C = S
so, v₂² = v₁² + 2aS [ use formula v² = u² + 2as , between B and C]
now, Let speed at D is v
distance between B and D = S/2
so, v² = v₁² + 2aS/2 = v₁² + aS ----(1)

Similarly in case of DC ,
v₂² = v² + 2aS/2 = v² + aS -----(2)

From (1) and (2), v² - v₁² = v₂² - v²
v ² = (v₁² + v₂²)/2
taking square root both sides,
v = √{(v₁² + v₂²)/2}