| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2001 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Inverse power force - gravitational/escape velocity context |
| Difficulty | Standard +0.8 This M3 question requires deriving a differential equation from acceleration (using chain rule v dv/dx = a), recognizing k = gR² from surface conditions, then solving by separation and applying boundary conditions. While methodical, it demands fluency with variable force mechanics, calculus techniques, and careful algebraic manipulation across multiple steps—significantly above average difficulty but standard for M3 material. |
| Spec | 6.06a Variable force: dv/dt or v*dv/dx methods |
A projectile $P$ is fired vertically upwards from a point on the earth's surface. When $P$ is at a distance $x$ from the centre of the earth its speed is $v$. Its acceleration is directed towards the centre of the earth and has magnitude $\frac{k}{x^2}$, where $k$ is a constant. The earth may be assumed to be a sphere of radius $R$.
\begin{enumerate}[label=(\alph*)]
\item Show that the motion of $P$ may be modelled by the differential equation
$$v \frac{dv}{dx} = -\frac{gR^2}{x^2}.$$ [4]
\end{enumerate}
The initial speed of $P$ is $U$, where $U^2 < 2gR$. The greatest distance of $P$ from the centre of the earth is $X$.
\begin{enumerate}[label=(\alph*)]\setcounter{enumi}{1}
\item Find $X$ in terms of $U$, $R$ and $g$. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2001 Q4 [10]}}