| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Topic | Momentum and Collisions 1 |
| Type | Three-particle sequential collisions |
| Difficulty | Standard +0.3 This is a standard sequential collision problem requiring systematic application of conservation of momentum and Newton's restitution law. Part (i) is routine algebra with given masses and coefficient, parts (ii-iii) involve substitution and repeating the method, and part (iv) requires simple comparison of velocities. The multi-step nature and careful bookkeeping elevate it slightly above average, but it follows a well-practiced template with no novel insights required. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.03i Coefficient of restitution: e6.03k Newton's experimental law: direct impact |
**Question 7**
**(i)** Let the velocities of $A$ and $B$ after the collision be $v$ and $w$.
$4mu = 4mv + 2mw$
$\therefore 2u = 2v + w$ — M1 (Use of conservation of momentum: a correct equation, consistent with a diagram, if present.)
$eu = w - v$ — M1 (Use of N.E.L.: a correct equation, consistent with a diagram, if present.)
$\therefore v = \frac{1}{3}(2-e)u$ and $w = \frac{2}{3}(1+e)u$ — M1, A1 **[4]**
Solve simultaneous equations. Both correct. Accept "$w$" unsimplified.
**(ii)** If $e = \frac{1}{2}$ then $v = \frac{1}{2}u$ and $w = u$ — B1 **[1]**
Ft *their* $v$ and $w$ in (i).
**(iii)** After $A$ collides with $B$ velocities are: $u/2$, $u$ (and $0$) respectively. — M1
After $B$ collides with $C$ velocities are: $u/2$, $u/2$ and $u$ respectively. — A1 **[2]**
Apply the result from (i) at least once. Or a complete correct method for the $BC$ collision. All correct, including $A$.
**(iv)** $A$ and $B$ have the same velocity and $C$ is moving away from them so there can be no further collisions. — B1 **[1]**
Ft (iii). Must consider all 3 particles.
**Total: [8]**7 A particle $A$ of mass $4 m$, on a smooth horizontal plane, is moving with speed $u$ directly towards another particle $B$, of mass $2 m$, which is at rest. The coefficient of restitution between the two particles is $e$.\\
(i) Show that, after the collision, the velocity of $A$ is $\frac { 1 } { 3 } ( 2 - e ) u$ and find the velocity of $B$.\\
(ii) Hence write down their velocities in the case when $e = \frac { 1 } { 2 }$.
Particle $B$ now collides directly with a third particle $C$, of mass $m$, which is at rest. The coefficient of restitution in both collisions is $\frac { 1 } { 2 }$.\\
(iii) Use your answers to part (ii) to find the velocities of $A , B$ and $C$ after the second collision has taken place.\\
(iv) Explain briefly whether any further collisions take place.
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2013 Q7 [8]}}