| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2007 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Impulse and momentum (advanced) |
| Type | Collision with fixed wall |
| Difficulty | Challenging +1.2 This is a standard M4 oblique impact question requiring resolution of velocities parallel and perpendicular to the wall, application of Newton's experimental law (e = v_perp_after/v_perp_before), and kinetic energy calculation. While it involves multiple steps and careful angle work, the method is entirely routine for M4 students who have practiced oblique impacts. The given angles lead to clean algebraic manipulation, making this a straightforward application rather than a problem requiring insight. |
| Spec | 6.02d Mechanical energy: KE and PE concepts6.03k Newton's experimental law: direct impact |
| Answer | Marks |
|---|---|
| \(u \cos 60° = v \cos 30°\) | M1A1 |
| \(u = v\sqrt{3}\) | A1 |
| \(KE \text{ lost} = \frac{1}{2}m(u^2 - v^2)\) | M1 |
| \(\text{Fraction of KE lost} = 1 - \left(\frac{v}{u}\right)^2\) | DM1 |
| \(= 1 - \frac{1}{3} = \frac{2}{3}\) or at least 3sf ending in 7 | A1 |
| or \(\frac{3}{4}(1 - e^2)\) | A1 |
| Total: 6 marks |
| Answer | Marks |
|---|---|
| \(e = \frac{v \sin 30°}{u \sin 60°}\) | M1A1 |
| \(= \frac{v}{u} \cdot \frac{1}{\sqrt{3}}\) | DM1 |
| \(= \frac{1}{3}\) | A1 |
| Total: 4 marks |
## Part (a)
| $u \cos 60° = v \cos 30°$ | M1A1 |
| $u = v\sqrt{3}$ | A1 |
| $KE \text{ lost} = \frac{1}{2}m(u^2 - v^2)$ | M1 |
| $\text{Fraction of KE lost} = 1 - \left(\frac{v}{u}\right)^2$ | DM1 |
| $= 1 - \frac{1}{3} = \frac{2}{3}$ or at least 3sf ending in 7 | A1 |
| or $\frac{3}{4}(1 - e^2)$ | A1 |
| **Total: 6 marks** | |
## Part (b)
| $e = \frac{v \sin 30°}{u \sin 60°}$ | M1A1 |
| $= \frac{v}{u} \cdot \frac{1}{\sqrt{3}}$ | DM1 |
| $= \frac{1}{3}$ | A1 |
| **Total: 4 marks** | |
### Guidance for both parts:
**a)** M1: Resolve parallel to the wall. Alt: reasonable attempt at equation connecting two variables. A1: Correct as above or equivalent equation correct. A1: $u$ in terms of $v$ or v.v. - not necessarily simplified, or ration of the two variables correct. M1: expression for KE lost. DM1: expression in one variable for fraction of KE lost – could be $u/v$ as above. A1: cao - accept decimals to at least 3sf.
**b)** M1: Use NIL perpendicular to the wall and form equation in $e$. A1: Correct unsimplified expression as above or $e u \sin 60° = v \sin 30°$ or equivalent. DM1: Substitute values for trig functions or use relationship from (a) and rearrange to $e = ...$. A1: cao - accept decimals to at least 3sf.
---A small ball is moving on a horizontal plane when it strikes a smooth vertical wall. The coefficient of restitution between the ball and the wall is $e$. Immediately before the impact the direction of motion of the ball makes an angle of $60°$ with the wall. Immediately after the impact the direction of motion of the ball makes an angle of $30°$ with the wall.
\begin{enumerate}[label=(\alph*)]
\item Find the fraction of the kinetic energy of the ball which is lost in the impact. [6]
\item Find the value of $e$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 2007 Q1 [10]}}