| Exam Board | OCR |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2006 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Momentum and Collisions 1 |
| Type | Collision with mass ratio parameter |
| Difficulty | Standard +0.3 This is a standard M2 collision question requiring conservation of momentum, Newton's experimental law (restitution), and kinetic energy calculations. Part (i) uses the physical constraint that sphere B cannot move backwards, part (ii) is direct application of the restitution formula, and part (iii) involves straightforward KE calculations. All techniques are routine for M2 students with no novel insight required, making it slightly easier than average. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(10 = 4 + m \cdot x\) | M1 | conservation of momentum |
| \(e = \ldots\) or rationale for \(x = 2\) | M1 | |
| \(m = 3\) | A1 | 3 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(v = 6\) | B1 | |
| \(e = 4/5\) or \(0.8\) | M1, A1 | 3 marks total; allow sign errors for M mark; watch out for lost minuses |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(10 - 5 = 2x + y\ (5 = -2a + b)\) | M1 | |
| \((-5 = 2c + d)\) | ||
| \(e = 0.8 = (y-x)/10\) | A1, M1 | look for consistency |
| \(y = x + 8 \quad (a + b = 8)\ (c - d = 8)\) | A1 | |
| \(x = -1\ (a=1)\ (c=1)\) | A1 | or 1 in opposite direction to 1st |
| \(y = 7\ (b=7)\ (d=-7)\) | A1 | |
| \(\frac{1}{2} \cdot 2.5^2 + \frac{1}{2} \cdot 1.5^2 - \frac{1}{2} \cdot 2.1^2 - \frac{1}{2} \cdot 1.7^2\) | M1 | K.E. lost. Must be 4 parts |
| \(12\) J | A1 | 8 marks total; \((37.5 - 25.5)\); 14 marks total |
# Question 8:
## Part (i):
| Working | Mark | Guidance |
|---------|------|----------|
| $10 = 4 + m \cdot x$ | M1 | conservation of momentum |
| $e = \ldots$ or rationale for $x = 2$ | M1 | |
| $m = 3$ | A1 | 3 marks total |
## Part (ii):
| Working | Mark | Guidance |
|---------|------|----------|
| $v = 6$ | B1 | |
| $e = 4/5$ or $0.8$ | M1, A1 | 3 marks total; allow sign errors for M mark; watch out for lost minuses |
## Part (iii):
| Working | Mark | Guidance |
|---------|------|----------|
| $10 - 5 = 2x + y\ (5 = -2a + b)$ | M1 | |
| $(-5 = 2c + d)$ | | |
| $e = 0.8 = (y-x)/10$ | A1, M1 | look for consistency |
| $y = x + 8 \quad (a + b = 8)\ (c - d = 8)$ | A1 | |
| $x = -1\ (a=1)\ (c=1)$ | A1 | or 1 in opposite direction to 1st |
| $y = 7\ (b=7)\ (d=-7)$ | A1 | |
| $\frac{1}{2} \cdot 2.5^2 + \frac{1}{2} \cdot 1.5^2 - \frac{1}{2} \cdot 2.1^2 - \frac{1}{2} \cdot 1.7^2$ | M1 | K.E. lost. Must be 4 parts |
| $12$ J | A1 | 8 marks total; $(37.5 - 25.5)$; 14 marks total |8 Two uniform smooth spheres, $A$ and $B$, have the same radius. The mass of $A$ is 2 kg and the mass of $B$ is $m \mathrm {~kg}$. Sphere $A$ is travelling in a straight line on a smooth horizontal surface, with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, when it collides directly with sphere $B$, which is at rest. As a result of the collision, sphere $A$ continues in the same direction with a speed of $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.\\
(i) Find the greatest possible value of $m$.
It is given that $m = 1$.\\
(ii) Find the coefficient of restitution between $A$ and $B$.
On another occasion $A$ and $B$ are travelling towards each other, each with speed $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, when they collide directly.\\
(iii) Find the kinetic energy lost due to the collision.
\hfill \mbox{\textit{OCR M2 2006 Q8 [14]}}