| Exam Board | WJEC |
|---|---|
| Module | Further Unit 3 (Further Unit 3) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Direct collision with given impulse |
| Difficulty | Standard +0.3 This is a standard Further Maths mechanics question on direct collisions requiring straightforward application of conservation of momentum, coefficient of restitution formula, and impulse-momentum theorem. All parts follow routine procedures with no conceptual surprises, though it requires careful sign conventions and multiple steps across the subparts. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03f Impulse-momentum: relation6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Conservation of momentum | M1 | Attempted. Allow 1 sign error |
| Answer | Marks | Guidance |
|---|---|---|
| \(v = 6 \cdot 5\) (ms\(^{-1}\)) | A1, A1[3] | All correct; Convincing |
| Answer | Marks | Guidance |
|---|---|---|
| Restitution | M1 | Attempted. Allow 1 sign error |
| Answer | Marks | Guidance |
|---|---|---|
| \(e = \frac{2}{5}\) | A1, A1[3] | All correct, oe; cao |
| Answer | Marks | Guidance |
|---|---|---|
| Change in momentum = 36 | M1 | |
| \((4m)(9 - 1 \cdot 5) = 36\) (30m = 36) | A1 | Correct equation, oe; or \((3m)(6 \cdot 5 - (-3 \cdot 5)) = 36\) |
| \(m = 1 \cdot 2\) | A1[3] | cao |
| Answer | Marks |
|---|---|
| Valid reason, eg. Radii are equal; Velocities are parallel to line of centres | E1[1] |
## Part (a)
Conservation of momentum | M1 | Attempted. Allow 1 sign error
$(9)(4m) + (-3 \cdot 5)(3m) = (1 \cdot 5)(4m) + (v)(3m)$
$25 \cdot 5 = 6 + 3v$
$v = 6 \cdot 5$ (ms$^{-1}$) | A1, A1[3] | All correct; Convincing
## Part (b)
Restitution | M1 | Attempted. Allow 1 sign error
$6 \cdot 5 - 1 \cdot 5 = -e(-3 \cdot 5 - 9)$
$5 = 12 \cdot 5e$
$e = \frac{2}{5}$ | A1, A1[3] | All correct, oe; cao
## Part (c)
Change in momentum = 36 | M1 |
$(4m)(9 - 1 \cdot 5) = 36$ (30m = 36) | A1 | Correct equation, oe; or $(3m)(6 \cdot 5 - (-3 \cdot 5)) = 36$
$m = 1 \cdot 2$ | A1[3] | cao
## Part (d)
Valid reason, eg. Radii are equal; Velocities are parallel to line of centres | E1[1] |
**Total for Question 3: 10**
---Two spheres $A$ and $B$, of equal radii, are moving towards each other on a smooth horizontal surface and collide directly. Sphere $A$ has mass $4m$ kg and sphere $B$ has mass $3m$ kg. Just before the collision, $A$ has speed $9\text{ ms}^{-1}$ and $B$ has speed $3.5\text{ ms}^{-1}$. Immediately after the collision, $A$ has speed $1.5\text{ ms}^{-1}$ in the direction of its original motion.
\begin{enumerate}[label=(\alph*)]
\item Show that the speed of $B$ immediately after the collision is $6.5\text{ ms}^{-1}$. [3]
\item Calculate the coefficient of restitution between $A$ and $B$. [3]
\item Given that the magnitude of the impulse exerted by $B$ on $A$ is 36 Ns, find the value of $m$. [3]
\item Give a reason why it is not necessary to model the spheres as particles in this question. [1]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 3 2022 Q3 [10]}}