| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2011 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Finding constants from motion conditions |
| Difficulty | Standard +0.3 This is a straightforward variable acceleration problem requiring integration to find displacement, solving v=0 for time at B, substituting into displacement equation, and differentiation to find maximum velocity. All steps are standard M1 techniques with no novel insight required, making it slightly easier than average. |
| Spec | 3.02f Non-uniform acceleration: using differentiation and integration3.02g Two-dimensional variable acceleration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | M1 | For using \(s = \int v \, dt\) |
| Displacement is \(2t^3 - kt^2/4\) | A1 | 2 |
| (ii) \(t = k/6\) | B1 | 1 |
| (iii) \([2 \times 216/k^3 - k \times 1296/4k^4 = 108 \to 2 \times 216 - 1296/4 = 108k^3]\) | dM1 | For substituting for t in displacement and equating to 108 |
| \(k = 1\) | A1 | 2 |
| (iv) \(dv/dt = 12t - 3kt^2\) | B1 | |
| \(= 0\) when \(t = (0), 4\) | B1 | |
| maximum value is 32 | B1 | 3 |
(i) | M1 | For using $s = \int v \, dt$
Displacement is $2t^3 - kt^2/4$ | A1 | 2
(ii) $t = k/6$ | B1 | 1
(iii) $[2 \times 216/k^3 - k \times 1296/4k^4 = 108 \to 2 \times 216 - 1296/4 = 108k^3]$ | dM1 | For substituting for t in displacement and equating to 108
$k = 1$ | A1 | 2
(iv) $dv/dt = 12t - 3kt^2$ | B1 |
$= 0$ when $t = (0), 4$ | B1 |
maximum value is 32 | B1 | 35 A particle $P$ moves in a straight line. It starts from rest at $A$ and comes to rest instantaneously at $B$. The velocity of $P$ at time $t$ seconds after leaving $A$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $v = 6 t ^ { 2 } - k t ^ { 3 }$ and $k$ is a constant.\\
(i) Find an expression for the displacement of $P$ from $A$ in terms of $t$ and $k$.\\
(ii) Find an expression for $t$ in terms of $k$ when $P$ is at $B$.
Given that the distance $A B$ is 108 m , find\\
(iii) the value of $k$,\\
(iv) the maximum value of $v$ when the particle is moving from $A$ towards $B$.
\hfill \mbox{\textit{CAIE M1 2011 Q5 [8]}}