| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2018 |
| Session | June |
| Marks | 13 |
| Topic | Oblique and successive collisions |
| Type | Particle bouncing on inclined plane |
| Difficulty | Challenging +1.8 This is a challenging oblique collision problem requiring resolution perpendicular/parallel to an inclined plane, impulse-momentum calculations with coefficient of restitution, and tracking successive bounces with projectile motion between impacts. It demands strong geometric visualization, systematic application of multiple mechanics principles, and careful multi-step calculation across three parts, placing it well above average difficulty but within reach of well-prepared Further Maths students. |
| Spec | 6.03f Impulse-momentum: relation6.03g Impulse in 2D: vector form6.03j Perfectly elastic/inelastic: collisions6.03k Newton's experimental law: direct impact |
| Answer | Marks | Guidance |
|---|---|---|
| - Impulse \(= 0.2(3--6) = 0.2(3+6)\) M1 \(0.2\times( | u_t | + |
| Answer | Marks | Guidance |
|---|---|---|
| - \(v_y = -3\), \(\therefore u'_y = 1.5\) M1 \( | u'_y | = |
**Question 11(i)**
- $U_y = -6$ soi **B1** $\perp$ component of $U$
- $\therefore u_y = 3$ **M1** $\pm0.5 \times$ perp component
- Impulse $= 0.2(3--6) = 0.2(3+6)$ **M1** $0.2\times(|u_t|+|U_y|)$
- $= 1.8$ Ns **A1** ignore units
- perpendicular to and away from the plane **B1**
**Question 11(ii)**
- $u_x = 8$, $a_y = (-)8$ **B1** both of these soi in **(ii)**
- $y = 3t - 4t^2$ (or $0 = 3 - 8\times0.5t$ or $-3 = 3 - 8t$) **M1** $y$-equation used, their $u_y$, $a_y$
- $= 0$ at $t = 0.75$ (or $t (= 2\times3/8) = 0.75$) **A1** obtaining $t = 0.75$ validly
- $x = 8t - 3t^2 = 4.3125$ or awrt 4.31 **A1** using $x$-equation to obtain $4\frac{5}{16}$ oe
**Question 11(iii)**
- $v_y = -3$, $\therefore u'_y = 1.5$ **M1** $|u'_y| = |0.5v_y|$
- $1.5t - 4t^2 = 0$ or $0 = 1.5 - 8\times0.5t$ or $-1.5 = 1.5 - 8t$ **M1** $y$-eqn., new $u_y$ (and old $a_y$)
- $t = 0.375$ **A1** finding new $t$
- Total time is $0.75 + 0.375 = 1.125$ cwo **A1** awrt 1.13 or $1\frac{1}{8}$11 A particle of mass 0.2 kg is projected so that it hits a smooth sloping plane $\Pi$ that makes an angle of $\sin ^ { - 1 } 0.6$ above the horizontal. The path of the particle is in a vertical plane containing a line of greatest slope of $\Pi$. Immediately before the first impact between the particle and $\Pi$, the particle is moving horizontally with speed $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The coefficient of restitution between the particle and $\Pi$ is 0.5 .\\
(i) Find the magnitude of the impulse on the particle from $\Pi$ at the first impact, and state the direction of this impulse.\\
(ii) Find the distance between the points on $\Pi$ where the first and second impacts occur.\\
(iii) Find the time taken between the first and third impacts.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2018 Q11 [13]}}