| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Verifying given motion properties |
| Difficulty | Standard +0.3 This is a straightforward variable acceleration question requiring standard calculus techniques: integration to find displacement, differentiation to find acceleration, and solving a quadratic equation. Part (i) involves verification by integration and simple differentiation. Part (ii) requires finding maximum velocity (setting dv/dt=0) then solving a quadratic, which are routine A-level mechanics procedures with no novel problem-solving required. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.08d Evaluate definite integrals: between limits3.02a Kinematics language: position, displacement, velocity, acceleration |
| Answer | Marks | Guidance |
|---|---|---|
| 6(i) | s = 0.3t2 – 0.01t3 | 1 |
| s(5) = 0.3 × 52 – 0.01 × 53 = 6.25 | 1 | A1 |
| a = 0.6 – 0.06t | 1 | M1 |
| a(5) = 0.6 – 0.0 × 5 = 0.3 ms–2 | 1 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 6(ii) | Maximum velocity is when | |
| 0.6 – 0.06t = 0 | 1 | M1 |
| [t = 10] | 1 | M1 |
| Max velocity = 3 ms–1 | 1 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| [t2 – 20t + 50 = 0] | 1 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Times are 2.93 s | 1 | A1 |
| and 17.07 s | 1 | A1 |
Question 6:
--- 6(i) ---
6(i) | s = 0.3t2 – 0.01t3 | 1 | M1 | For integration
s(5) = 0.3 × 52 – 0.01 × 53 = 6.25 | 1 | A1
a = 0.6 – 0.06t | 1 | M1 | For differentiation
a(5) = 0.6 – 0.0 × 5 = 0.3 ms–2 | 1 | A1
4
--- 6(ii) ---
6(ii) | Maximum velocity is when
0.6 – 0.06t = 0 | 1 | M1 | For setting a = 0
[t = 10] | 1 | M1 | For solving a = 0
Max velocity = 3 ms–1 | 1 | A1
0.6t – 0.03t2 = 1.5
[t2 – 20t + 50 = 0] | 1 | M1 | Setting velocity = half its maximum and attempting to
solve a three term quadratic
Times are 2.93 s | 1 | A1
and 17.07 s | 1 | A1
6A particle $P$ moves in a straight line, starting from a point $O$. The velocity of $P$, measured in m s$^{-1}$, at time $t$ s after leaving $O$ is given by
$$v = 0.6t - 0.03t^2.$$
\begin{enumerate}[label=(\roman*)]
\item Verify that, when $t = 5$, the particle is 6.25 m from $O$. Find the acceleration of the particle at this time. [4]
\item Find the values of $t$ at which the particle is travelling at half of its maximum velocity. [6]
\end{enumerate}
\hfill \mbox{\textit{CAIE M1 Q6 [10]}}