| Exam Board | OCR MEI |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2013 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Variable mass problems |
| Type | Sand/mass accumulation on vehicle |
| Difficulty | Challenging +1.2 This is a standard variable mass problem from M4/Further Mechanics requiring application of momentum conservation with dm/dt = k. Part (i) involves setting up and solving a separable differential equation, then integrating to find displacement - both routine techniques for this topic. Part (ii) is straightforward substitution. While this requires more sophistication than basic mechanics, it's a textbook application of variable mass theory without novel insight. |
| Spec | 6.03a Linear momentum: p = mv6.03b Conservation of momentum: 1D two particles |
1 An empty railway truck of mass $m _ { 0 }$ is moving along a straight horizontal track at speed $v _ { 0 }$. The point P is at the front of the truck. The horizontal forces on the truck are negligible. As P passes a fixed point O , sand starts to fall vertically into the truck at a constant mass rate $k$. At time $t$ after P passes O the speed of the truck is $v$ and $\mathrm { OP } = x$.\\
(i) Find an expression for $v$ in terms of $m _ { 0 } , v _ { 0 } , k$ and $t$, and show that $x = \frac { m _ { 0 } v _ { 0 } } { k } \ln \left( 1 + \frac { k t } { m _ { 0 } } \right)$.\\
(ii) Find the speed of the truck and the distance OP when the mass of sand in the truck is $2 m _ { 0 }$.
\hfill \mbox{\textit{OCR MEI M4 2013 Q1 [11]}}