| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2018 |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Advanced work-energy problems |
| Type | Inverse-square gravitational force |
| Difficulty | Standard +0.3 This is a standard M3 gravitational force question requiring (a) equating the inverse square law to weight at Earth's surface (routine substitution), and (b) applying conservation of energy with variable gravitational force. While it involves integration or the work-energy principle, this is a textbook application of core M3 content with no novel insight required, making it slightly easier than average. |
| Spec | 3.03a Force: vector nature and diagrams6.02i Conservation of energy: mechanical energy principle |
| VIIIV SIHI NI JIIIM ION OC | VIIV SIHI NI JAHM ION OO | VI4V SIHI NI JIIIM I ON OO |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(F = \frac{K}{x^2}\) | — | Starting point |
| \(x = R \Rightarrow F = mg \therefore mg = \frac{K}{R^2}\) | M1 | Setting \(F = mg\) and \(x = R\) |
| \(K = mgR^2\) | A1 | Deducing the given answer |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\frac{mgR^2}{x^2} = -mv\frac{dv}{dx}\) | M1 | Attempting equation of motion with acceleration in form \(v\frac{dv}{dx}\); minus sign may be missing |
| \(g\int \frac{R^2}{x^2}dx = -\int v\,dv\) | — | Separating variables |
| \(-g\frac{R^2}{x} = -\frac{1}{2}v^2 \quad (+c)\) | dM1, A1ft | Attempting integration; correct integration, follow through on missing minus sign, constant of integration may be missing |
| \(x = 3R,\; v = V \Rightarrow -g\frac{R^2}{3R} = -\frac{1}{2}V^2 + c\) | M1 | Substituting \(x = 3R, v = V\) to obtain equation for \(c\) |
| \(c = -\frac{Rg}{3} + \frac{1}{2}V^2\) | A1 | Correct expression for \(c\) |
| \(x = R \Rightarrow \frac{1}{2}v^2 = -\frac{Rg}{3} + \frac{1}{2}V^2 + g\frac{R^2}{R}\) | M1 | Substituting \(x = R\) and their expression for \(c\) |
| \(v^2 = V^2 + \frac{4Rg}{3}\) | — | Simplification |
| \(v = \sqrt{V^2 + \frac{4Rg}{3}}\) | A1 cso | Correct expression for \(v\), any equivalent form |
## Question 2:
### Part (a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $F = \frac{K}{x^2}$ | — | Starting point |
| $x = R \Rightarrow F = mg \therefore mg = \frac{K}{R^2}$ | M1 | Setting $F = mg$ **and** $x = R$ |
| $K = mgR^2$ | A1 | Deducing the given answer |
**(2 marks)**
### Part (b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\frac{mgR^2}{x^2} = -mv\frac{dv}{dx}$ | M1 | Attempting equation of motion with acceleration in form $v\frac{dv}{dx}$; minus sign may be missing |
| $g\int \frac{R^2}{x^2}dx = -\int v\,dv$ | — | Separating variables |
| $-g\frac{R^2}{x} = -\frac{1}{2}v^2 \quad (+c)$ | dM1, A1ft | Attempting integration; correct integration, follow through on missing minus sign, constant of integration may be missing |
| $x = 3R,\; v = V \Rightarrow -g\frac{R^2}{3R} = -\frac{1}{2}V^2 + c$ | M1 | Substituting $x = 3R, v = V$ to obtain equation for $c$ |
| $c = -\frac{Rg}{3} + \frac{1}{2}V^2$ | A1 | Correct expression for $c$ |
| $x = R \Rightarrow \frac{1}{2}v^2 = -\frac{Rg}{3} + \frac{1}{2}V^2 + g\frac{R^2}{R}$ | M1 | Substituting $x = R$ and their expression for $c$ |
| $v^2 = V^2 + \frac{4Rg}{3}$ | — | Simplification |
| $v = \sqrt{V^2 + \frac{4Rg}{3}}$ | A1 cso | Correct expression for $v$, any equivalent form |
**(7 marks) — Total: 9 marks**
---2. A spacecraft $S$ of mass $m$ moves in a straight line towards the centre of the Earth. The Earth is modelled as a sphere of radius $R$ and $S$ is modelled as a particle. When $S$ is at a distance $x , x \geqslant R$, from the centre of the Earth, the force exerted by the Earth on $S$ is directed towards the centre of the Earth. The force has magnitude $\frac { K } { x ^ { 2 } }$, where $K$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that $K = m g R ^ { 2 }$
When $S$ is at a distance $3 R$ from the centre of the Earth, the speed of $S$ is $V$. Assuming that air resistance can be ignored,
\item find, in terms of $g , R$ and $V$, the speed of $S$ as it hits the surface of the Earth.
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VIIIV SIHI NI JIIIM ION OC & VIIV SIHI NI JAHM ION OO & VI4V SIHI NI JIIIM I ON OO \\
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\end{center}
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\hfill \mbox{\textit{Edexcel M3 2018 Q2 [9]}}