| Exam Board | OCR |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2012 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Oblique collision, find velocities/angles |
| Difficulty | Standard +0.3 This is a standard M3 oblique collision problem requiring conservation of momentum along the line of centres and Newton's restitution law. The symmetry that makes B's speed unchanged is elegant but emerges naturally from the setup. Requires multiple steps but uses routine techniques with no novel insight needed—slightly easier than average A-level. |
| Spec | 6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2mu\cos\alpha - mu\cos\alpha = 2ma + mb\) | M1 | For using the p.c.m. parallel to l.o.c. Allow sign errors, \(m/2m\), \(\sin/\cos\) |
| \(0.5(u\cos\alpha + u\cos\alpha) = b - a\) | M1 | For using NEL parallel to l.o.c. Allow sign errors, \(e\) left in |
| A1 | For both p.c.m. and NEL correct & consistent, dep on M1M1 gained | |
| Comp of \(B\)'s velocity along l.o.c. is \(u\cos\alpha\) | A1ft | By stating vel perp l.o.c. still \(u\sin\alpha\), hence result, dep on all previous marks |
| Establishing \(B\)'s speed unchanged | A1 | Or by showing speed is still \(u\); condone 'vertical' in this part |
| [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a = 0\) | B1 | May be shown in (i) |
| Correct interpretation of direction of \(A\) | B1 | Perp to l.o.c.; condone 'vertical', accept sketch |
| Direction of \(B\) is at angle \(\alpha\) to l.o.c., with an indication that removes ambiguity (e.g. sketch) | B1 | Refs to sketch in (i) |
| [3] |
## Question 2:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2mu\cos\alpha - mu\cos\alpha = 2ma + mb$ | M1 | For using the p.c.m. parallel to l.o.c. Allow sign errors, $m/2m$, $\sin/\cos$ |
| $0.5(u\cos\alpha + u\cos\alpha) = b - a$ | M1 | For using NEL parallel to l.o.c. Allow sign errors, $e$ left in |
| | A1 | For both p.c.m. and NEL correct & consistent, dep on M1M1 gained |
| Comp of $B$'s velocity along l.o.c. is $u\cos\alpha$ | A1ft | By stating vel perp l.o.c. still $u\sin\alpha$, hence result, dep on all previous marks |
| Establishing $B$'s speed unchanged | A1 | Or by showing speed is still $u$; condone 'vertical' in this part |
| **[5]** | | |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a = 0$ | B1 | May be shown in (i) |
| Correct interpretation of direction of $A$ | B1 | Perp to l.o.c.; condone 'vertical', accept sketch |
| Direction of $B$ is at angle $\alpha$ to l.o.c., with an indication that removes ambiguity (e.g. sketch) | B1 | Refs to sketch in (i) |
| **[3]** | | |
---2\\
\includegraphics[max width=\textwidth, alt={}, center]{43ed8ec7-67f1-418a-8d4e-ee96448647fd-2_544_816_781_603}
Two uniform smooth spheres $A$ and $B$, of equal radius, have masses $2 m \mathrm {~kg}$ and $m \mathrm {~kg}$ respectively. They are moving in opposite directions on a horizontal surface and they collide. Immediately before the collision, each sphere has speed $u \mathrm {~ms} ^ { - 1 }$ in a direction making an angle $\alpha$ with the line of centres (see diagram). The coefficient of restitution between $A$ and $B$ is 0.5 .\\
(i) Show that the speed of $B$ is unchanged as a result of the collision.\\
(ii) Find the direction of motion of each of the spheres after the collision.
\hfill \mbox{\textit{OCR M3 2012 Q2 [8]}}