| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2020 |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Piecewise motion functions |
| Difficulty | Moderate -0.3 This is a standard piecewise velocity function question requiring continuity to find k, sketching, identifying where acceleration is positive (v' > 0), and integration for distance. All techniques are routine M1 material with straightforward calculus, making it slightly easier than average A-level. |
| Spec | 3.02f Non-uniform acceleration: using differentiation and integration3.02g Two-dimensional variable acceleration |
| Answer | Marks | Guidance |
|---|---|---|
| 6(a) | \(k = 40\) | B1 |
| 6(b) | Correct for \(0 \leq x < 4\) For horizontal line at \(y = 20\); for \(4 \leq x \leq 20\) (20, 28); \(y = 4\); \(y = 14\) to \(y = 20\) | BFT BFT BFT |
| 6(c) | For attempting to differentiate to find \(a = 10 - 10\) Allow < 4 but not 1 | M1 A1 |
| 6(d) | \(\text{Dist} = \left[\frac{5t^2}{s^2} - \frac{100t}{3} + \frac{5 \times (40-10)}{2}\right] + (40 - 10) + (0.5 \times (40 + 28) \times 6)\) \(= 64\text{m}\) | A1 |
**6(a)** | $k = 40$ | B1 |
**6(b)** | Correct for $0 \leq x < 4$ For horizontal line at $y = 20$; for $4 \leq x \leq 20$ (20, 28); $y = 4$; $y = 14$ to $y = 20$ | BFT BFT BFT | Correct for 10 ≤ $x$ ≤ 14; FT on $k$
**6(c)** | For attempting to differentiate to find $a = 10 - 10$ Allow < 4 but not 1 | M1 A1 | For correct integration
**6(d)** | $\text{Dist} = \left[\frac{5t^2}{s^2} - \frac{100t}{3} + \frac{5 \times (40-10)}{2}\right] + (40 - 10) + (0.5 \times (40 + 28) \times 6)$ $= 64\text{m}$ | A1 |
---6 A particle $P$ moves in a straight line. The velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time $t \mathrm {~s}$ is given by
$$\begin{array} { l l }
v = 5 t ( t - 2 ) & \text { for } 0 \leqslant t \leqslant 4 \\
v = k & \text { for } 4 \leqslant t \leqslant 14 \\
v = 68 - 2 t & \text { for } 14 \leqslant t \leqslant 20
\end{array}$$
where $k$ is a constant.\\
(a) Find $k$.\\
(b) Sketch the velocity-time graph for $0 \leqslant t \leqslant 20$.\\
(c) Find the set of values of $t$ for which the acceleration of $P$ is positive.\\
(d) Find the total distance travelled by $P$ in the interval $0 \leqslant t \leqslant 20$.\\
\hfill \mbox{\textit{CAIE M1 2020 Q6 [11]}}