# [SOLVED]frictionless railcar

#### dwsmith

##### Well-known member
Two hobos, each of mass $m_{\text{h}}$, are standing at one end of a stationary railroad flatcar with frictionless wheels and mass $m_{\text{fc}}$.
Either hobo can run to the other end of the flatcar and jump off with the same speed $u$ (relative to the car).

Use conservation of momentum to find the speed of the recoiling car if the two men run and jump off simultaneously.

Let $v$ be the velocity of the recoiling car.
Then
\begin{alignat*}{3}
m_{\text{fc}}v & = & 2m_{\text{h}}(u - v)\\
v & = & \frac{2m_{\text{h}}}{m_{\text{fc}}}(u - v)
\end{alignat*}

The solution is $v = \frac{2m_{\text{h}}}{2m_{\text{h}}+m_{\text{fc}}}u$.
How did they get that?

#### Fantini

MHB Math Helper
Well, expanding the product gets us

$$m_{fc} v = 2m_h (u-v) = 2m_h u - 2m_h v,$$

and from that

$$m_{fc} v + 2m_h v = (m_{fc} + 2m_h) v = 2m_h u.$$

Finally,

$$v = \frac{2m_h}{m_{fc} + 2m_h} u.$$

If you don't mind me asking, is this question from the book Classical Mechanics by John Taylor? Cheers.

#### dwsmith

##### Well-known member
It is from that book but I figured out what they did before you posted; hence, the post was marked solved before then.