| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2021 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Finding when particle at rest |
| Difficulty | Moderate -0.3 This is a straightforward kinematics question requiring standard techniques: factorizing a quadratic to find when v=0, sketching a parabola, and integrating velocity (with attention to sign changes) to find distance. While it requires multiple steps and careful handling of the direction change, all techniques are routine M1 content with no novel problem-solving required. |
| Spec | 1.08a Fundamental theorem of calculus: integration as reverse of differentiation3.02a Kinematics language: position, displacement, velocity, acceleration3.02b Kinematic graphs: displacement-time and velocity-time |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((2t-3)(t-1) = 0\) leading to \(t = \ldots\) | M1 | Attempt to solve \(v = 0\) |
| \(t = 1\) or \(t = 1.5\) | A1 | |
| Minimum velocity when \(t = 1.25\) leading to \(v = \ldots\); or \(\dfrac{dv}{dt} = 4t - 5 = 0,\ t = 1.25\) leading to \(v = \ldots\); or \(v = 2\left[\left(t - \dfrac{5}{4}\right)^2 - \dfrac{25}{16}\right] + 3\) leading to \(v = \ldots\) | M1 | Uses roots or \(dv/dt = 0\) to find \(t\) for \(v_{\min}\) and attempts substitution to obtain \(v_{\min}\). Alternatively completes square |
| Minimum velocity is \(-0.125\ \text{m s}^{-1}\) | A1 | Allow \(v = -\dfrac{1}{8}\) |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Quadratic curve (two roots and \(v(3) > v(0)\)) | B1 | |
| Goes through \((1.25,\ -0.125),\ (0,\ 3),\ (1,\ 0),\ (1.5,\ 0),\ (3,\ 6)\) | B1 | 3 of the 5 key points shown on axes or as coordinates |
| All five points shown on a totally correct graph | B1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(s = \dfrac{2}{3}t^3 - \dfrac{5}{2}t^2 + 3t\) | M1 | For use of \(s = \int v\, dt\) |
| \(\left[\dfrac{2}{3}(1.5)^3 - \dfrac{5}{2}(1.5)^2 + 3(1.5)\right] - \left[\dfrac{2}{3}(1)^3 - \dfrac{5}{2}(1)^2 + 3(1)\right]\) | M1 | Correct use of limits (their 1 and 1.5) |
| Distance \(= 0.0417\) m | A1 | A0 for \(-0.0417\) |
| 3 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(2t-3)(t-1) = 0$ leading to $t = \ldots$ | M1 | Attempt to solve $v = 0$ |
| $t = 1$ or $t = 1.5$ | A1 | |
| Minimum velocity when $t = 1.25$ leading to $v = \ldots$; or $\dfrac{dv}{dt} = 4t - 5 = 0,\ t = 1.25$ leading to $v = \ldots$; or $v = 2\left[\left(t - \dfrac{5}{4}\right)^2 - \dfrac{25}{16}\right] + 3$ leading to $v = \ldots$ | M1 | Uses roots or $dv/dt = 0$ to find $t$ for $v_{\min}$ and attempts substitution to obtain $v_{\min}$. Alternatively completes square |
| Minimum velocity is $-0.125\ \text{m s}^{-1}$ | A1 | Allow $v = -\dfrac{1}{8}$ |
| | **4** | |
---
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Quadratic curve (two roots and $v(3) > v(0)$) | B1 | |
| Goes through $(1.25,\ -0.125),\ (0,\ 3),\ (1,\ 0),\ (1.5,\ 0),\ (3,\ 6)$ | B1 | 3 of the 5 key points shown on axes or as coordinates |
| All five points shown on a totally correct graph | B1 | |
| | **3** | |
---
## Question 6(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $s = \dfrac{2}{3}t^3 - \dfrac{5}{2}t^2 + 3t$ | M1 | For use of $s = \int v\, dt$ |
| $\left[\dfrac{2}{3}(1.5)^3 - \dfrac{5}{2}(1.5)^2 + 3(1.5)\right] - \left[\dfrac{2}{3}(1)^3 - \dfrac{5}{2}(1)^2 + 3(1)\right]$ | M1 | Correct use of limits (their 1 and 1.5) |
| Distance $= 0.0417$ m | A1 | A0 for $-0.0417$ |
| | **3** | |
---6 A particle moves in a straight line and passes through the point $A$ at time $t = 0$. The velocity of the particle at time $t \mathrm {~s}$ after leaving $A$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where
$$v = 2 t ^ { 2 } - 5 t + 3$$
\begin{enumerate}[label=(\alph*)]
\item Find the times at which the particle is instantaneously at rest. Hence or otherwise find the minimum velocity of the particle.
\item Sketch the velocity-time graph for the first 3 seconds of motion.
\item Find the distance travelled between the two times when the particle is instantaneously at rest.
\end{enumerate}
\hfill \mbox{\textit{CAIE M1 2021 Q6 [10]}}