| Exam Board | OCR |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2007 |
| Session | June |
| Marks | 16 |
| 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, coefficient of restitution, and analysis of conditions for a second collision. While the individual techniques (momentum conservation, Newton's experimental law) are standard M2 content, part (iii) requires careful reasoning about relative velocities and inequalities, and part (v) involves a second collision calculation. The problem demands sustained multi-step reasoning across five parts, elevating it above routine M2 exercises but not requiring truly novel insights. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03f Impulse-momentum: relation6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(1.8 = -0.3 + 3m\) | M1 | |
| \(m = 0.7\) | A1 2 | AG |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(e = 4/6\) | M1 | accept \(2/6\) for M1 |
| \(2/3\) | A1 2 | accept \(0.67\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\pm 3f\) | B1 | |
| \(\frac{1}{3} \diamond f\ (\bigotimes\ 1)\) | B1 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(I = 3f \times 0.7 - -3 \times 0.7\) | M1 | ok for only one minus sign for M1 |
| A1 | ||
| \(I = 2.1(f+1)\) | A1 3 | aef — 2 marks only for \(-2.1(f+1)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(0.3 + 6.3/4 = 0.3a + 0.7b\) | M1 | can be \(-0.7b\) |
| \(3a + 7b = 18.75\) | A1 \(*\) | aef |
| \(\frac{2}{3} = \frac{(a-b)/5}{4}\) | M1 | allow \(e=3/4\) or their \(e\) for M1 |
| \(3a - 3b = 5/2\) | A1 \(*\) | aef — \(*\) means dependent |
| solve | M1 | |
| \(a = 2.5\) | A1 | \((2.46)\) allow \(\pm\ (59/24)\) |
| \(b = 1.6\) | A1 7 | \((1.625)\) allow \(\pm\ (13/8)\) |
## Question 7(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $1.8 = -0.3 + 3m$ | M1 | |
| $m = 0.7$ | A1 **2** | **AG** |
## Question 7(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $e = 4/6$ | M1 | accept $2/6$ for M1 |
| $2/3$ | A1 **2** | accept $0.67$ |
## Question 7(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\pm 3f$ | B1 | |
| $\frac{1}{3} \diamond f\ (\bigotimes\ 1)$ | B1 **2** | |
## Question 7(iv):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $I = 3f \times 0.7 - -3 \times 0.7$ | M1 | ok for only one minus sign for M1 |
| | A1 | |
| $I = 2.1(f+1)$ | A1 **3** | aef — 2 marks only for $-2.1(f+1)$ |
## Question 7(v):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $0.3 + 6.3/4 = 0.3a + 0.7b$ | M1 | can be $-0.7b$ |
| $3a + 7b = 18.75$ | A1 $*$ | aef |
| $\frac{2}{3} = \frac{(a-b)/5}{4}$ | M1 | allow $e=3/4$ or their $e$ for M1 |
| $3a - 3b = 5/2$ | A1 $*$ | aef — $*$ means dependent |
| solve | M1 | |
| $a = 2.5$ | A1 | $(2.46)$ allow $\pm\ (59/24)$ |
| $b = 1.6$ | A1 **7** | $(1.625)$ allow $\pm\ (13/8)$ |
---7 Two small spheres $A$ and $B$, with masses 0.3 kg and $m \mathrm {~kg}$ respectively, lie at rest on a smooth horizontal surface. $A$ is projected directly towards $B$ with speed $6 \mathrm {~ms} ^ { - 1 }$ and hits $B$. The direction of motion of $A$ is reversed in the collision. The speeds of $A$ and $B$ after the collision are $1 \mathrm {~ms} ^ { - 1 }$ and $3 \mathrm {~ms} ^ { - 1 }$ respectively. The coefficient of restitution between $A$ and $B$ is $e$.\\
(i) Show that $m = 0.7$.\\
(ii) Find $e$.
B continues to move at $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and strikes a vertical wall at right angles. The coefficient of restitution between $B$ and the wall is $f$.\\
(iii) Find the range of values of $f$ for which there will be a second collision between $A$ and $B$.\\
(iv) Find, in terms of $f$, the magnitude of the impulse that the wall exerts on $B$.\\
(v) Given that $f = \frac { 3 } { 4 }$, calculate the final speeds of $A$ and $B$, correct to 1 decimal place.
\hfill \mbox{\textit{OCR M2 2007 Q7 [16]}}