| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2007 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Inverse power force - gravitational/escape velocity context |
| Difficulty | Standard +0.8 Part (a) is straightforward substitution using Newton's law at Earth's surface. Part (b) requires setting up and integrating the equation of motion F=ma with variable force, or applying work-energy principles with variable force over distance—a multi-step M3-level problem requiring careful handling of limits and integration, but following standard variable force methodology. |
| Spec | 3.03d Newton's second law: 2D vectors6.02i Conservation of energy: mechanical energy principle6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks | Guidance |
|---|---|---|
| (a) At surface: \(\frac{k}{R^2} = mg \Rightarrow k = mgR^2\) | M1 A1 (cso) | 2 marks |
| (b) N2L: \(m\ddot{x} = -\frac{mgR^2}{x^2}\) | ||
| \(v\frac{dv}{dx} = -\frac{gR^2}{x^2}\) or \(\frac{d}{dx}\left(\frac{1}{2}v^2\right) = -\frac{gR^2}{x^2}\) | M1; M1 | |
| \(\int v dv = -gR^2 \int \frac{1}{x^2}dx\) or \(\frac{1}{2}v^2 = -gR^2 \int \frac{1}{x^2}dx\) | M1; M1 | |
| \(\frac{1}{2}v^2 = \frac{gR^2}{x} (+C)\) | A1 | |
| \(x = 2R, v = 0 \Rightarrow C = -\frac{gR}{2}\) | M1 A1 | |
| \(v^2 = \frac{2gR^2}{x} - gR\) | ||
| At \(x = R\): \(v^2 = \frac{2gR^2}{R} - gR = 2gR - gR\) | M1 | |
| \(v = \sqrt{(gR)}\) | A1 | 7 marks [9] |
**(a)** At surface: $\frac{k}{R^2} = mg \Rightarrow k = mgR^2$ | M1 A1 (cso) | 2 marks
**(b)** N2L: $m\ddot{x} = -\frac{mgR^2}{x^2}$ | |
$v\frac{dv}{dx} = -\frac{gR^2}{x^2}$ or $\frac{d}{dx}\left(\frac{1}{2}v^2\right) = -\frac{gR^2}{x^2}$ | M1; M1 |
$\int v dv = -gR^2 \int \frac{1}{x^2}dx$ or $\frac{1}{2}v^2 = -gR^2 \int \frac{1}{x^2}dx$ | M1; M1 |
$\frac{1}{2}v^2 = \frac{gR^2}{x} (+C)$ | A1 |
$x = 2R, v = 0 \Rightarrow C = -\frac{gR}{2}$ | M1 A1 |
$v^2 = \frac{2gR^2}{x} - gR$ | |
At $x = R$: $v^2 = \frac{2gR^2}{R} - gR = 2gR - gR$ | M1 |
$v = \sqrt{(gR)}$ | A1 | 7 marks [9]
---A spacecraft $S$ of mass $m$ moves in a straight line towards the centre of the earth. The earth is modelled as a fixed sphere of radius $R$. When $S$ is at a distance $x$ from the centre of the earth, the force exerted by the earth on $S$ is directed towards the centre of the earth and has magnitude $\frac{k}{x^2}$, where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that $k = mgR^2$. [2]
\end{enumerate}
Given that $S$ starts from rest when its distance from the centre of the earth is $2R$, and that air resistance can be ignored,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the speed of $S$ as it crashes into the surface of the earth. [7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2007 Q3 [9]}}