Question

A $10,000kg$railroad car is rolling at $2.0m/s$ when a $4000kg$ load of gravel is suddenly dropped in. What is the car’s speed just after the gravel is loaded?

Short answer

The speed of the car just after the gravel is loaded is$1.4m/s$.

Step-by-step solution

Given information    

We need to find the speed of the car just after the gravel is loaded.

Simplify  

When the initial momentum equals the final momentum, the momentum of an isolated system changes to zero. The particle interactions are isolated from the external environment. Because momentum is conserved, we apply the law of conservation of momentum, which is given by equation in the form

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

Because of isolated system the railway is frictionless. The object has mass $m$and moves with speed $v$that has momentum $p$, Vector is a product of object's mass and its velocity.The momentum is given by equation in the form

$p=mv$

Using expression $p$into equation $\left(1\right)$

$m_f{v}_f=m_i{v}_i\left(2\right)$

A car's initial mass is $m_i=10000kg$, and the final mass is the car's mass plus the mass of the gravel added$m_f=10000kg+4000kg=14000kg$. Initial speed is $v_i=2.0m/s$.The final speed should be found. So, after solving equation $\left(2\right)$for $v_f$

$$\begin{gathered} v_f=\frac{m_i{v}_i}{m_f} \\ =\frac{\left(10000kg\right)\left(2.0m/s\right)}{140000kg} \\ =1.4m/s \end{gathered}$$