| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2017 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Successive collisions, three particles in line |
| Difficulty | Challenging +1.8 This is a challenging Further Maths mechanics problem requiring multiple collision calculations with coefficient of restitution, algebraic manipulation to find an inequality constraint, and verification that no further collisions occur. It demands systematic application of conservation of momentum and Newton's law of restitution across successive collisions, plus careful analysis of velocity conditions—significantly harder than standard A-level mechanics but follows established collision methodology. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(mv_A + kmv_B = 2mu + kmu\); \([v_A + kv_B = 2u + ku]\) | M1 | Use conservation of momentum for \(A\) & \(B\) (allow omission of \(m\) in all momentum equations) |
| \(v_B - v_A = \frac{1}{2}(2u - u)\ [= \frac{1}{2}u]\) | M1 | Use Newton's restitution law with consistent LHS signs |
| \(v_B = u(2k+5)/2(k+1)\ \text{or}\ u(k+5/2)/(k+1)\) | A1 | Combine to find \(v_B\) |
| \([v_A = u(k+4)/2(k+1)]\ v_B > 4u/3\ \text{if}\ k < 7/2\) | M1 A1 | Find inequality for \(k\) from speeds of \(B\) and \(C\) after 1st collision |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(kmw_B + mv_C = kmv_B + m(4u/3)\); \([2w_B + v_C = 2(3u/2) + 4u/3 = 13u/3\ \text{when}\ k=2]\) | M1 | Use conservation of momentum for \(B\) & \(C\) |
| \(v_C - w_B = \frac{1}{2}(v_B - 4u/3)\ [= u/12]\); \((k+1)w_B = (k - \frac{1}{2})v_B + 2u\) | M1 | Use Newton's restitution law with consistent LHS signs; Combine to find \(w_B\) |
| \(3w_B = (3/2)v_B + 2u\) with \(v_B = 3u/2\), so \(w_B = 17u/12\) | *A1 | when \(k = 2\) |
| \(v_A = u,\ v_A < w_B\) | DB1 | Verify no further collisions between \(A\) and \(B\) |
| *EITHER:* \((k+1)v_C = (3k/2)v_B + (2-k)(2u/3)\); \(3v_C = 3v_B\) with \(v_B = 3u/2\) so \(v_C = 3u/2 > w_B\) | (DB1) | *EITHER:* Find \(v_C\) and verify no further collisions between \(B\) and \(C\) |
| *OR:* \(B\) and \(C\) cannot meet again since they move apart after colliding | (DB1) | *OR:* State explicitly that no further collisions between \(B\) and \(C\) |
| 5 |
# Question 3(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $mv_A + kmv_B = 2mu + kmu$; $[v_A + kv_B = 2u + ku]$ | **M1** | Use conservation of momentum for $A$ & $B$ (allow omission of $m$ in all momentum equations) |
| $v_B - v_A = \frac{1}{2}(2u - u)\ [= \frac{1}{2}u]$ | **M1** | Use Newton's restitution law with consistent LHS signs |
| $v_B = u(2k+5)/2(k+1)\ \text{or}\ u(k+5/2)/(k+1)$ | **A1** | Combine to find $v_B$ |
| $[v_A = u(k+4)/2(k+1)]\ v_B > 4u/3\ \text{if}\ k < 7/2$ | **M1 A1** | Find inequality for $k$ from speeds of $B$ and $C$ after 1st collision |
| | **5** | |
# Question 3(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $kmw_B + mv_C = kmv_B + m(4u/3)$; $[2w_B + v_C = 2(3u/2) + 4u/3 = 13u/3\ \text{when}\ k=2]$ | **M1** | Use conservation of momentum for $B$ & $C$ |
| $v_C - w_B = \frac{1}{2}(v_B - 4u/3)\ [= u/12]$; $(k+1)w_B = (k - \frac{1}{2})v_B + 2u$ | **M1** | Use Newton's restitution law with consistent LHS signs; Combine to find $w_B$ |
| $3w_B = (3/2)v_B + 2u$ with $v_B = 3u/2$, so $w_B = 17u/12$ | ***A1** | when $k = 2$ |
| $v_A = u,\ v_A < w_B$ | **DB1** | Verify no further collisions between $A$ and $B$ |
| *EITHER:* $(k+1)v_C = (3k/2)v_B + (2-k)(2u/3)$; $3v_C = 3v_B$ with $v_B = 3u/2$ so $v_C = 3u/2 > w_B$ | **(DB1)** | *EITHER:* Find $v_C$ and verify no further collisions between $B$ and $C$ |
| *OR:* $B$ and $C$ cannot meet again since they move apart after colliding | **(DB1)** | *OR:* State explicitly that no further collisions between $B$ and $C$ |
| | **5** | |3 Three uniform small smooth spheres $A , B$ and $C$ have equal radii and masses $m , k m$ and $m$ respectively, where $k$ is a constant. The spheres are moving in the same direction along a straight line on a smooth horizontal surface, with $B$ between $A$ and $C$. The speeds of $A , B$ and $C$ are $2 u , u$ and $\frac { 4 } { 3 } u$ respectively. The coefficient of restitution between any pair of the spheres is $\frac { 1 } { 2 }$. After sphere $A$ has collided with sphere $B$, sphere $B$ collides with sphere $C$.\\
(i) Find an inequality satisfied by $k$.\\
(ii) Given that $k = 2$, show that after $B$ has collided with $C$ there are no further collisions between any of the three spheres.\\
\hfill \mbox{\textit{CAIE FP2 2017 Q3 [10]}}