| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Successive collisions with wall rebound |
| Difficulty | Standard +0.8 This is a multi-stage collision problem requiring conservation of momentum, Newton's restitution law, and careful tracking of velocities through three collision events. While the techniques are standard for Further Maths mechanics, the problem requires sustained algebraic manipulation across multiple parts, with part (iii) demanding insight into whether a second collision occurs based on relative velocities. This is more challenging than typical A-level mechanics but not exceptionally difficult for Further Maths students. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| \(4mv_A + mv_B = 4mu\) (AEF) | M1 | Use momentum (allow \(m\) omitted) |
| \(v_B - v_A = eu\) | M1 | Use Newton's law (M0 if LHS signs inconsistent) |
| \(v_A = \frac{1}{5}(4 - e)\,u\) (AG) | A1 | Combine to verify/find speeds of \(A\) and \(B\) after collision |
| \(v_B = \frac{4}{5}(1 + e)\,u\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(v_B' = [-]\,\frac{3}{4}e\,v_B \left[= -\frac{3}{5}e(1+e)\,u\right]\) | M1 | Relate vel. \(v_B'\) of \(B\) after colln. with wall to \(v_B\) |
| \(\frac{1}{5}(4-e)\,u = \frac{3}{5}e(1+e)\,u\) | M1 | Equate speeds of \(A\) and \(B\) (ignore sign of \(v_B'\) for both M1s) |
| \(3e^2 + 4e - 4 = 0,\; e = \frac{2}{3}\) | A1 | Solve resulting quadratic for \(e\), [implicitly] rejecting root \(-2\) |
| Answer | Marks | Guidance |
|---|---|---|
| *EITHER:* \(4mw_A + mw_B = 3mv_A \left[= 2mu\right]\) (AEF) | M1 A1 | Use momentum (allow \(m\) omitted) with \(v_A = -v_B' \left[= \frac{2}{3}u\right]\); Use Newton's law (M0 if LHS signs inconsistent); Combine to find velocity of \(B\) after final collision |
| \(w_B - w_A = e \times 2v_A = \frac{4}{3}v_A\) or \(\frac{8}{9}u\) (AEF) | Allow any similar valid argument | |
| \(\left[w_A = \frac{1}{3}v_A = \frac{2}{9}u\right],\; w_B = \frac{5}{3}v_A\) or \(\frac{10}{9}u\) | Allow \(v_A > 0\) by [implicit] inspection | |
| *OR:* Momentum before colln. is \(3mv_A = 2mu > 0\), so after colln. momentum *or* speed of \(B > 0\); \(w_B > 0\) so \(B\) collides again with barrier |
**Question 2(i):**
$4mv_A + mv_B = 4mu$ (AEF) | M1 | Use momentum (allow $m$ omitted)
$v_B - v_A = eu$ | M1 | Use Newton's law (M0 if LHS signs inconsistent)
$v_A = \frac{1}{5}(4 - e)\,u$ (AG) | A1 | Combine to verify/find speeds of $A$ and $B$ after collision
$v_B = \frac{4}{5}(1 + e)\,u$ | A1 |
**Total: 4**
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**Question 2(ii):**
$v_B' = [-]\,\frac{3}{4}e\,v_B \left[= -\frac{3}{5}e(1+e)\,u\right]$ | M1 | Relate vel. $v_B'$ of $B$ after colln. with wall to $v_B$
$\frac{1}{5}(4-e)\,u = \frac{3}{5}e(1+e)\,u$ | M1 | Equate speeds of $A$ and $B$ (ignore sign of $v_B'$ for both M1s)
$3e^2 + 4e - 4 = 0,\; e = \frac{2}{3}$ | A1 | Solve resulting quadratic for $e$, [implicitly] rejecting root $-2$
**Total: 3**
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**Question 2(iii):**
*EITHER:* $4mw_A + mw_B = 3mv_A \left[= 2mu\right]$ (AEF) | M1 A1 | Use momentum (allow $m$ omitted) with $v_A = -v_B' \left[= \frac{2}{3}u\right]$; Use Newton's law (M0 if LHS signs inconsistent); Combine to find velocity of $B$ after final collision
$w_B - w_A = e \times 2v_A = \frac{4}{3}v_A$ or $\frac{8}{9}u$ (AEF) | | Allow any similar valid argument
$\left[w_A = \frac{1}{3}v_A = \frac{2}{9}u\right],\; w_B = \frac{5}{3}v_A$ or $\frac{10}{9}u$ | | Allow $v_A > 0$ by [implicit] inspection
*OR:* Momentum before colln. is $3mv_A = 2mu > 0$, so after colln. momentum *or* speed of $B > 0$; $w_B > 0$ so $B$ collides again with barrier | |
**Total: 2**2 Two uniform small spheres $A$ and $B$ have equal radii and masses $4 m$ and $m$ respectively. Sphere $A$ is moving with speed $u$ on a smooth horizontal surface when it collides directly with sphere $B$ which is at rest. The coefficient of restitution between the spheres is $e$.\\
(i) Show that after the collision $A$ moves with speed $\frac { 1 } { 5 } u ( 4 - e )$ and find the speed of $B$.\\
Sphere $B$ continues to move until it collides with a fixed smooth vertical barrier which is perpendicular to the direction of motion of $B$. The coefficient of restitution between $B$ and the barrier is $\frac { 3 } { 4 } e$. After this collision, the speeds of $A$ and $B$ are equal.\\
(ii) Find the value of $e$.\\
The spheres $A$ and $B$ now collide directly again.\\
(iii) Determine whether sphere $B$ collides with the barrier for a second time.\\
\hfill \mbox{\textit{CAIE FP2 2018 Q2 [9]}}