| Exam Board | OCR |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Oblique and successive collisions |
| Type | Oblique collision, direction deflected given angle |
| Difficulty | Standard +0.8 This is a multi-part oblique collision problem requiring resolution of velocities parallel and perpendicular to the line of centres, conservation of momentum, energy loss calculations, and coefficient of restitution. While the individual techniques are standard M3 content, the question requires careful bookkeeping across four interconnected parts with given numerical constraints that must be satisfied simultaneously. This is moderately harder than a typical mechanics question but not exceptionally challenging for M3 level. |
| Spec | 6.03a Linear momentum: p = mv6.03b Conservation of momentum: 1D two particles6.03c Momentum in 2D: vector form6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact6.03l Newton's law: oblique impacts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{2} \times 2(5^2 - v^2) = 7.56 \quad (v^2 = 17.44)\) | M1, A1 | For using \(\frac{1}{2}m(u^2 - v^2) = 7.56\) and solving for \(v\); must use '5', allow sign error, missing \(\frac{1}{2}\), missing \(m\) |
| Speed is 4.18 ms\(^{-1}\) | A1 | Do not award if this is not candidate's final answer |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(v_{Ay} = u_{Ay} = 5\sin\alpha = 4\) | B1 | |
| \([v_{Ax}^2 + 4^2 = 17.44 \Rightarrow v_{Ax}^2 = 1.44]\) | M1 | For using \(v_{Ax}^2 + v_{Ay}^2 = 17.44\) |
| \(v_{Ax} = \pm 1.2\) and \(v_{Ax}\) must be less than 0.8 \(\Rightarrow\) component has magnitude 1.2 ms\(^{-1}\) and direction to the left | A1 | |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2 \times 3 - m \times 2 = 2 \times (-1.2) + m \times 0.8\) | M1, A1 FT | For using the pcm parallel to loc; must use \(5\cos\alpha\), 2, 0.8 and '1.2', 4 terms or equivalent; allow sign errors; condone one mass missing. FT incorrect \(v_{AX}\) |
| \(m = 3\) | A1 | CAO |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([e(3+2) = (1.2 + 0.8)]\) | M1 | For using NEL with their '1.2' and \(5\cos\alpha\), 2 and 0.8; allow sign errors. Must be right way up |
| \(e = 0.4\) | A1 | |
| [2] |
# Question 6:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{2} \times 2(5^2 - v^2) = 7.56 \quad (v^2 = 17.44)$ | M1, A1 | For using $\frac{1}{2}m(u^2 - v^2) = 7.56$ and solving for $v$; must use '5', allow sign error, missing $\frac{1}{2}$, missing $m$ |
| Speed is 4.18 ms$^{-1}$ | A1 | Do not award if this is not candidate's final answer |
| | **[3]** | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $v_{Ay} = u_{Ay} = 5\sin\alpha = 4$ | B1 | |
| $[v_{Ax}^2 + 4^2 = 17.44 \Rightarrow v_{Ax}^2 = 1.44]$ | M1 | For using $v_{Ax}^2 + v_{Ay}^2 = 17.44$ |
| $v_{Ax} = \pm 1.2$ and $v_{Ax}$ must be less than 0.8 $\Rightarrow$ component has magnitude 1.2 ms$^{-1}$ and direction to the left | A1 | |
| | **[3]** | |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2 \times 3 - m \times 2 = 2 \times (-1.2) + m \times 0.8$ | M1, A1 FT | For using the pcm parallel to loc; must use $5\cos\alpha$, 2, 0.8 and '1.2', 4 terms or equivalent; allow sign errors; condone one mass missing. FT incorrect $v_{AX}$ |
| $m = 3$ | A1 | CAO |
| | **[3]** | |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[e(3+2) = (1.2 + 0.8)]$ | M1 | For using NEL with their '1.2' and $5\cos\alpha$, 2 and 0.8; allow sign errors. Must be right way up |
| $e = 0.4$ | A1 | |
| | **[2]** | |6\\
\includegraphics[max width=\textwidth, alt={}, center]{cc74a925-f1c8-4f59-a421-b46444cae5ec-5_387_867_258_575}
Two smooth uniform spheres $A$ and $B$, of equal radius, have masses 2 kg and $m \mathrm {~kg}$ respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, $A$ has speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and is moving towards $B$ at an angle of $\alpha$ to the line of centres, where $\cos \alpha = 0.6$. $B$ has speed $2 \mathrm {~ms} ^ { - 1 }$ and is moving towards $A$ along the line of centres (see diagram). As a result of the collision, $A$ 's loss of kinetic energy is $7.56 \mathrm {~J} , B$ 's direction of motion is reversed and $B$ 's speed after the collision is $0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find\\
(i) the speed of $A$ after the collision,\\
(ii) the component of $A$ 's velocity after the collision, parallel to the line of centres, stating with a reason whether its direction is to the left or to the right,\\
(iii) the value of $m$,\\
(iv) the coefficient of restitution between $A$ and $B$.\\
$7 S _ { A }$ and $S _ { B }$ are light elastic strings. $S _ { A }$ has natural length 2 m and modulus of elasticity $120 \mathrm {~N} ; S _ { B }$ has natural length 3 m and modulus of elasticity 180 N . A particle $P$ of mass 0.8 kg is attached to one end of each of the strings. The other ends of $S _ { A }$ and $S _ { B }$ are attached to fixed points $A$ and $B$ respectively, on a smooth horizontal table. The distance $A B$ is $6 \mathrm {~m} . P$ is released from rest at the point of the line segment $A B$ which is 2.9 m from $A$.\\
(i) For the subsequent motion, show that the total elastic potential energy of the strings is the same when $A P = 2.1 \mathrm {~m}$ and when $A P = 2.9 \mathrm {~m}$. Deduce that neither string becomes slack.\\
(ii) Find, in terms of $x$, an expression for the acceleration of $P$ in the direction of $A B$ when $A P = ( 2.5 + x ) \mathrm { m }$.\\
(iii) State, giving a reason, the type of motion of $P$ and find the time taken between successive occasions when $P$ is instantaneously at rest.
For the instant 0.6 seconds after $P$ is released, find\\
(iv) the distance travelled by $P$,\\
(v) the speed of $P$.
\hfill \mbox{\textit{OCR M3 2012 Q6 [11]}}