| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2012 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Displacement from velocity by integration |
| Difficulty | Moderate -0.8 This is a straightforward mechanics question requiring basic differentiation to find acceleration, solving a linear equation, and integration of a polynomial to find displacement. All steps are routine applications of standard techniques with no conceptual challenges or problem-solving insight required. |
| Spec | 3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \([a = 0.2 + 0.012t]\) | M1 | For differentiating to find \(a(t)\) |
| \([0.2 + 0.012t = 2.5 \times 0.2]\) | M1 | For attempting to solve \(a(t) = 2.5a(0)\) |
| \(t = 25\) | A1 | 3 marks total |
| (ii) \([s = 0.1t^2 + 0.002t^3 \text{ (+ C)}]\) | M1 | For integrating to find \(s(t)\) |
| \([s = 0.1 \times 625 + 0.002 \times 15625]\) | DM1 | For using limits 0 to 25 or evaluating \(s(t)\) with \(C = 0\) (which may be implied by its absence) |
| Displacement is 93.75 (accept 93.7 or 93.8) | A1 | 3 marks total |
**(i)** $[a = 0.2 + 0.012t]$ | M1 | For differentiating to find $a(t)$
$[0.2 + 0.012t = 2.5 \times 0.2]$ | M1 | For attempting to solve $a(t) = 2.5a(0)$
$t = 25$ | A1 | 3 marks total | AG
**(ii)** $[s = 0.1t^2 + 0.002t^3 \text{ (+ C)}]$ | M1 | For integrating to find $s(t)$
$[s = 0.1 \times 625 + 0.002 \times 15625]$ | DM1 | For using limits 0 to 25 or evaluating $s(t)$ with $C = 0$ (which may be implied by its absence)
Displacement is 93.75 (accept 93.7 or 93.8) | A1 | 3 marks total
---2 A particle moves in a straight line. Its velocity $t$ seconds after leaving a fixed point $O$ on the line is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $v = 0.2 t + 0.006 t ^ { 2 }$. For the instant when the acceleration of the particle is 2.5 times its initial acceleration,\\
(i) show that $t = 25$,\\
(ii) find the displacement of the particle from $O$.
\hfill \mbox{\textit{CAIE M1 2012 Q2 [6]}}