| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2021 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Piecewise motion functions |
| Difficulty | Standard +0.3 This is a standard M1 mechanics question requiring routine techniques: finding constants from continuity conditions, sketching a velocity-time graph, and calculating distance via integration. The connected particles problem uses standard pulley mechanics with energy considerations. All methods are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.08d Evaluate definite integrals: between limits3.02f Non-uniform acceleration: using differentiation and integration3.03l Newton's third law: extend to situations requiring force resolution3.03o Advanced connected particles: and pulleys |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(a = 2pt - q\) | *M1 | Attempt to differentiate \(v\) |
| \(36p - 6q = 36\), \(4p - q = 0\) | DM1 | For attempting to set up 2 equations using \(a = 0\) at \(t = 2\) and matching velocities at \(t = 6\), solve for \(p\) or \(q\) |
| \(p = 3, q = 12\) | A1 | Both correct |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Correct quadratic from \(t = 0\) to \(t = 6\), or correct straight line from 6 to 14 | B1 | No labelling necessary |
| Both quadratic and straight line correct | B1 | Must join; no labelling needed |
| All correct and key points shown | B1 | All correct, labelled at \((4, 0)\), \((6, 36)\) and \((14, 0)\) |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempt to integrate \(v\) | *M1 | Allow in terms of \(p\) and \(q\) |
| \(s = t^3 - 6t^2\) | A1 FT | FT on their \(p\) and \(q\) values |
| \(s(\text{quadratic}) = \left[t^3 - 6t^2\right]_0^4 + \left[t^3 - 6t^2\right]_4^6\) | DM1 | \([= 32 + 32]\); using limits correctly for \(t = 0\) to \(t = 6\); allow in terms of \(p\) and \(q\) |
| \(s(\text{triangle}) = \left[63t - 2.25t^2\right]_6^{14} = 144\), or area of triangle \(= 144\) | B1 | |
| Total distance travelled in \(14\text{ s} = 208\text{ m}\) | A1 | |
| [5] |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $a = 2pt - q$ | ***M1** | Attempt to differentiate $v$ |
| $36p - 6q = 36$, $4p - q = 0$ | **DM1** | For attempting to set up 2 equations using $a = 0$ at $t = 2$ and matching velocities at $t = 6$, solve for $p$ or $q$ |
| $p = 3, q = 12$ | **A1** | Both correct |
| | **[3]** | |
---
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Correct quadratic from $t = 0$ to $t = 6$, **or** correct straight line from 6 to 14 | **B1** | No labelling necessary |
| Both quadratic and straight line correct | **B1** | Must join; no labelling needed |
| All correct and key points shown | **B1** | All correct, labelled at $(4, 0)$, $(6, 36)$ and $(14, 0)$ |
| | **[3]** | |
---
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt to integrate $v$ | ***M1** | Allow in terms of $p$ and $q$ |
| $s = t^3 - 6t^2$ | **A1 FT** | FT on their $p$ and $q$ values |
| $s(\text{quadratic}) = \left[t^3 - 6t^2\right]_0^4 + \left[t^3 - 6t^2\right]_4^6$ | **DM1** | $[= 32 + 32]$; using limits correctly for $t = 0$ to $t = 6$; allow in terms of $p$ and $q$ |
| $s(\text{triangle}) = \left[63t - 2.25t^2\right]_6^{14} = 144$, or area of triangle $= 144$ | **B1** | |
| Total distance travelled in $14\text{ s} = 208\text{ m}$ | **A1** | |
| | **[5]** | |
---6 A particle $P$ moves in a straight line starting from a point $O$ and comes to rest 14 s later. At time $t \mathrm {~s}$ after leaving $O$, the velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ of $P$ is given by
$$\begin{array} { l l }
v = p t ^ { 2 } - q t & 0 \leqslant t \leqslant 6 \\
v = 63 - 4.5 t & 6 \leqslant t \leqslant 14
\end{array}$$
where $p$ and $q$ are positive constants.\\
The acceleration of $P$ is zero when $t = 2$.
\begin{enumerate}[label=(\alph*)]
\item Given that there are no instantaneous changes in velocity, find $p$ and $q$.
\item Sketch the velocity-time graph.
\item Find the total distance travelled by $P$ during the 14 s .\\
\includegraphics[max width=\textwidth, alt={}, center]{e1b91e54-a3ae-436c-a4f7-7095891f7034-10_326_1109_255_520}
Two particles $A$ and $B$ of masses 2 kg and 3 kg respectively are connected by a light inextensible string. Particle $B$ is on a smooth fixed plane which is at an angle of $18 ^ { \circ }$ to horizontal ground. The string passes over a fixed smooth pulley at the top of the plane. Particle $A$ hangs vertically below the pulley and is 0.45 m above the ground (see diagram). The system is released from rest with the string taut. When $A$ reaches the ground, the string breaks.
Find the total distance travelled by $B$ before coming to instantaneous rest. You may assume that $B$ does not reach the pulley.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE M1 2021 Q6 [11]}}