Question

Two ends of a train moving with a constant acceleration pass a certain point with velocities u and v. What is the velocity with which the middle point of the train passes the same point?

Answer 1

Solve.

Two ends of a train moving with a constant acceleration pass a certain point with velocities $u$ and $v$ . What is the velocity with which the middle point of the train passes the same point?

Solution.

Let $P$ and $Q$ be the two ends of the train and $O$ is the midpoint of $PQ$ . Let $PQ = L$ the length of the train.

When the end $Q$ crosses a certain point say $R$ , the velocity of every point of the train = velocity of $Q = u$ .

Hence, at the instant, the velocity of $O =$ velocity of $P = u$ .

When the end $P$ comes to $R$ its velocity $= v$ .

For the motion of $\mathsf{P}$ :

$$v ^ {2} = u ^ {2} + 2 a L$$

where $a = \text{const. acceleration}$

When the point $O$ comes to $R$ , its velocity $w$ is given by

$$w ^ {2} = u ^ {2} + 2 a \left(\frac {L}{2}\right) = u ^ {2} + a L$$

From (1) and (2):

$$a L = \frac {v ^ {2} - u ^ {2}}{2}$$ $$w ^ {2} = u ^ {2} + \left(\frac {v ^ {2} - u ^ {2}}{2}\right) = \frac {v ^ {2} + u ^ {2}}{2}$$

Or

$$w = \sqrt {\frac {v ^ {2} + u ^ {2}}{2}}$$

Answer:

$$w = \sqrt {\frac {v ^ {2} + u ^ {2}}{2}}$$