Solution to Question 36
1. System Definition
This is a variable mass problem. We consider the system consisting of the freight train and the coal collected inside it.
- Initial mass of empty train: $m_0$
- Rate of coal loading: $r$ (kg/s)
- Total mass at time $t$: $m(t) = m_0 + rt$
- Applied constant force: $F$
2. Applying Newton’s Second Law
For a variable mass system where mass is being added, the external force is equal to the rate of change of momentum:
$$ F_{ext} = \frac{dp}{dt} = \frac{d}{dt} (mv) $$
Here, the coal falls vertically from a stationary hopper. This means the coal has zero horizontal velocity just before it hits the train. Therefore, the force $F$ is responsible for accelerating both the train and the newly added mass.
3. Integration
We integrate the force equation with respect to time:
$$ \int_{0}^{t} F \, dt = \int_{0}^{v} d(mv) $$
Since $F$ is constant and the train starts from rest ($v=0$ at $t=0$):
$$ Ft = m(t)v(t) – m(0)v(0) $$
$$ Ft = (m_0 + rt)v – 0 $$
4. Final Expression
Rearranging the equation to solve for velocity $v$:
$$ v = \frac{Ft}{m_0 + rt} $$