| Exam Board | Edexcel |
|---|---|
| Module | M5 (Mechanics 5) |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Topic | Variable mass problems |
| Type | Body collecting atmospheric moisture |
| Difficulty | Challenging +1.8 This M5 question requires deriving a differential equation using variable mass mechanics (momentum principle with mass accretion), then solving a first-order linear ODE with an integrating factor. The variable mass aspect and the need to carefully apply F=dp/dt (not F=ma) makes this significantly harder than standard mechanics problems, requiring sophisticated mathematical technique and physical insight into non-standard situations. |
| Spec | 4.10c Integrating factor: first order equations6.03f Impulse-momentum: relation |
A raindrop falls vertically under gravity through a cloud. In a model of the motion the raindrop is assumed to be spherical at all times and the cloud is assumed to consist of stationary water particles. At time $t = 0$, the raindrop is at rest and has radius $a$. As the raindrop falls, water particles from the cloud condense onto it and the radius of the raindrop is assumed to increase at a constant rate $\lambda$. At time $t$ the speed of the raindrop is $v$.
\begin{enumerate}[label=(\alph*)]
\item Show that
$$\frac{dv}{dt} + \frac{3\lambda v}{(\lambda t + a)} = g.$$
[8]
\end{enumerate}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the speed of the raindrop when its radius is $3a$.
[7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M5 Q5 [15]}}