Question

the two ends of train moving with constant acceleration pass a certain point with velocity u and 3u. The velocity with which the middle point of the train passes the same point is

Answer 1

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Train is a rigid body i.e relative velocity between any two points of the train is always zero. And since we are assuming that train is rigid body, this implies train is accelerating so that the velocity of two ends can vary. Another assumption, this acceleration is always constant.

Let “L” m be the total length of the train. And let it be accelerating with a constant acceleration of “a” m/s²

Let the velocity of train as midpoint passes through the given point be “v” m/s

Given that when the starting point of train passes through the point, it has a speed of “u” m/s

When the end point of the train passes, it has a speed of “3u” m/s

We can assume the train to be a single particle which is moving with a constant acceleration whose intitial velocity is u, final velocity is 3u, distance travelled is L. We need to find the velocity of the particle when it travels a distance of L/2.

Formula used : V² - U² = 2as

Let's first find the acceleration of the train.

Substituting the values of V and U and s, we get,

9u² - u² = 2aL

a = 4u²/L

Now, velocity of particle at s = L/2

v² - u² = 2.a.L/2

v² = u² + L.4u²/L

v² = 5u²

v = √5u