Three identical train cars, coupled together, are rolling east at speed $v_o$. A fourth car traveling east at $2v_o$ catches up with the three and couples to make a four-car train. A moment later, the train cars hit a fifth car that was at rest on the tracks, and it couples to make a five-car train. What is the speed of the five-car train?

We need to find the speed of the five-car train?

When the initial momentum equals the final momentum, the momentum of an isolated system changes to zero. There is no interaction with the environment within the system.The momentum is conserved, and we therefore use equation (11.23) in the form of conservation of momentum

${\overrightarrow{p}}_f={\overrightarrow{p}}_i\left(1\right)$

A mass and speed object has momentum when they move with speed, and momentum is a vector, which is the product of mass and velocity.

$P=mv\left(2\right)$

First case:

To move with velocity $v_o$that coupled with three identical cars and same mass car moves with velocity $2v_o$the initial momentum of the system is

$p_i=\left(3m\right)\left(v_o\right)+\left(m\right)\left(2v_o\right)=5m{v}_o$

The four cars are coupled and moved with velocity $v_f$, $1$ after collision and momentum after collision is

$P_f=4m{v}_{f,1}$

Using equation $\left(1\right)$for expresion of $v_f$,$1$in the form

$$\begin{gathered} {\overrightarrow{p}}_f={\overrightarrow{p}}_f \\ 4m{v}_{f,1}=5m{v}_o \\ {v}_{f,1}=\frac{5}{4}{v}_o \\ {v_{f,}}_1=1.25v_o \end{gathered}$$

The four cars coupled and moves with velocity $1.25v_o$and the fifth cars is at rest $v_i,2=0$.Before collision the initial momentum of the system is

$p_i=\left(4m\right)(1.25v_o)+\left(m\right)\left(0\right)=5m{v}_o$

The five cars are coupled and moved with velocity $v_{f,2}$ after collision and momentum after collision is

$p_f=5m{v}_{f,2}$

Using equation $\left(1\right)$for expresion of $v_{f,2}$in the form

$$\begin{gathered} {\overrightarrow{P}}_f={\overrightarrow{P}}_i \\ 5m{v}_{f,2}=5m{v}_o \\ {v}_{f,2}=v_o \end{gathered}$$

Hence, fifth car's velocity is$v_o$.