| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2013 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Friction |
| Type | Projectile on rough surface |
| Difficulty | Standard +0.3 This is a standard two-particle kinematics problem with friction. Part (i) requires straightforward application of F=ma with friction (F=μR=μmg), giving constant decelerations. Part (ii) involves setting up and solving SUVAT equations for both particles to find when they meet, then calculating their speeds at collision. While it requires careful bookkeeping of two particles and multiple steps, the techniques are routine M1 content with no novel problem-solving insight needed. |
| Spec | 3.02d Constant acceleration: SUVAT formulae3.03v Motion on rough surface: including inclined planes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(-\mu mg = ma\) | M1 | For using Newton's 2nd law, \(F = \mu R\) and \(R = mg\) |
| Decelerations of P and Q are \(2\ \text{ms}^{-2}\) and \(2.5\ \text{ms}^{-2}\) | A1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| M1 | For using \(s = ut + \frac{1}{2}at^2\) and \(s_P = s_Q + 5\) | |
| \(8t - t^2 = 3t - 1.25t^2 + 5\) | A1 | |
| \(t = \sqrt{120} - 10\) (\(= 0.95445...\)) | A1 | |
| M1 | For using \(v = u + at\) for both P and Q | |
| Speed of P \(= 6.09\ \text{ms}^{-1}\), speed of Q \(= 0.614\ \text{ms}^{-1}\) | A1 [5] |
## Question 4:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $-\mu mg = ma$ | M1 | For using Newton's 2nd law, $F = \mu R$ and $R = mg$ |
| Decelerations of P and Q are $2\ \text{ms}^{-2}$ and $2.5\ \text{ms}^{-2}$ | A1 [2] | |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| | M1 | For using $s = ut + \frac{1}{2}at^2$ and $s_P = s_Q + 5$ |
| $8t - t^2 = 3t - 1.25t^2 + 5$ | A1 | |
| $t = \sqrt{120} - 10$ ($= 0.95445...$) | A1 | |
| | M1 | For using $v = u + at$ for both P and Q |
| Speed of P $= 6.09\ \text{ms}^{-1}$, speed of Q $= 0.614\ \text{ms}^{-1}$ | A1 [5] | |
---4 Particles $P$ and $Q$ are moving in a straight line on a rough horizontal plane. The frictional forces are the only horizontal forces acting on the particles.\\
(i) Find the deceleration of each of the particles given that the coefficient of friction between $P$ and the plane is 0.2 , and between $Q$ and the plane is 0.25 .
At a certain instant, $P$ passes through the point $A$ and $Q$ passes through the point $B$. The distance $A B$ is 5 m . The velocities of $P$ and $Q$ at $A$ and $B$ are $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, respectively, both in the direction $A B$.\\
(ii) Find the speeds of $P$ and $Q$ immediately before they collide.
\hfill \mbox{\textit{CAIE M1 2013 Q4 [7]}}