| Exam Board | OCR |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2014 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Direct collision, find velocities |
| Difficulty | Challenging +1.8 This M3 question combines circular motion energy methods with collision mechanics requiring conservation of momentum and restitution equations. While the setup is multi-stage (energy to find pre-collision speeds, collision analysis, then circular motion forces), each component uses standard techniques. The symmetry simplifies the collision to one dimension, and the given answer guides part (i). More demanding than typical A-level due to the M3 content and multi-step nature, but follows predictable patterns without requiring novel insight. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03i Coefficient of restitution: e6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact6.05d Variable speed circles: energy methods6.05e Radial/tangential acceleration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{2}mv^2 = mg\times 0.8(1-\sin 30°)\); \(v = 2.8\ \text{ms}^{-1}\) | M1, A1 | Or with '\(5m\)' if for \(Q\); allow \(g\) missing for M1; might see \(v^2 = 0.8g\) |
| Speed of P and Q equal; use conservation of momentum: \(5mx2.8 - mx2.8 = 5mq + mp\) | B1ft, B1ft | soi; ft on velocity |
| Use NEL: \(p - q = -0.95(-2.8-2.8)\); \(p = 6.3\ \text{ms}^{-1}\); \(q = 0.98\ \text{ms}^{-1}\), \(Q\) moves to left | M1, A1ft, A1, A1 [8] | Ft on velocity; supporting work required for AG; direction must be clear; \(p\) is vel of \(P\), \(q\) is vel of \(Q\), both to left; allow \(\pm e\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| By energy for \(P\) at top: \(\frac{1}{2}m6.3^2 = \frac{1}{2}mv^2 + mg\times 1.6\); \(v^2 = 8.33\) | M1, A1, A1 | Must have 3 terms; soi; allow \(g\) missing, sign error |
| Use \(F = ma\) at top: \(mg + R = m\times\frac{8.33}{0.8}\); \(R = 0.6125m\) | M1, A1ft, A1CAO [6] | Must have 3 terms; their \(v^2\); Or \(49m/80\); allow \(g\) missing, sign error |
## Question 6:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{2}mv^2 = mg\times 0.8(1-\sin 30°)$; $v = 2.8\ \text{ms}^{-1}$ | M1, A1 | Or with '$5m$' if for $Q$; allow $g$ missing for M1; might see $v^2 = 0.8g$ |
| Speed of P and Q equal; use conservation of momentum: $5mx2.8 - mx2.8 = 5mq + mp$ | B1ft, B1ft | soi; ft on velocity |
| Use NEL: $p - q = -0.95(-2.8-2.8)$; $p = 6.3\ \text{ms}^{-1}$; $q = 0.98\ \text{ms}^{-1}$, $Q$ moves to left | M1, A1ft, A1, A1 [8] | Ft on velocity; supporting work required for AG; direction must be clear; $p$ is vel of $P$, $q$ is vel of $Q$, both to left; allow $\pm e$ |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| By energy for $P$ at top: $\frac{1}{2}m6.3^2 = \frac{1}{2}mv^2 + mg\times 1.6$; $v^2 = 8.33$ | M1, A1, A1 | Must have 3 terms; soi; allow $g$ missing, sign error |
| Use $F = ma$ at top: $mg + R = m\times\frac{8.33}{0.8}$; $R = 0.6125m$ | M1, A1ft, A1CAO [6] | Must have 3 terms; their $v^2$; Or $49m/80$; allow $g$ missing, sign error |
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\includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-4_547_515_267_772}
A hollow cylinder is fixed with its axis horizontal. $O$ is the centre of a vertical cross-section of the cylinder and $D$ is the highest point on the cross-section. $A$ and $C$ are points on the circumference of the cross-section such that $A O$ and $C O$ are both inclined at an angle of $30 ^ { \circ }$ below the horizontal diameter through $O$. The inner surface of the cylinder is smooth and has radius 0.8 m (see diagram). A particle $P$, of mass $m \mathrm {~kg}$, and a particle $Q$, of mass $5 m \mathrm {~kg}$, are simultaneously released from rest from $A$ and $C$, respectively, inside the cylinder. $P$ and $Q$ collide; the coefficient of restitution between them is 0.95 .\\
(i) Show that, immediately after the collision, $P$ moves with speed $6.3 \mathrm {~ms} ^ { - 1 }$, and find the speed and direction of motion of $Q$.\\
(ii) Find, in terms of $m$, an expression for the normal reaction acting on $P$ when it subsequently passes through $D$.
\hfill \mbox{\textit{OCR M3 2014 Q6 [14]}}