| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2006 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Acceleration from velocity differentiation |
| Difficulty | Moderate -0.8 This is a straightforward mechanics question requiring basic differentiation to find acceleration and integration to find displacement. Both parts involve standard calculus techniques with simple polynomial functions, making it easier than average for A-level. The only mild challenge is coordinating the two parts, but the methods are routine textbook exercises. |
| Spec | 1.08d Evaluate definite integrals: between limits3.02f Non-uniform acceleration: using differentiation and integration3.02g Two-dimensional variable acceleration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(a(t) = 1.25 - 0.1t\) | B1 | May be scored in (ii) |
| Initial acceleration is 1.25\(ms^{-2}\) | B1 | 2 |
| (ii) \([t = 12]\) | M1 | For attempting to solve \(dv/dt = 0.05\) |
| \(1.25t^2/2 - 0.05t^3\) \((+C)\) | M1, A1 | For attempting to integrate \(v(t)\) |
| \([1.25 \times 12^2/2 - 0.05 \times 12^3/3 = 90 - 28.8]\) | DM1 | For using appropriate limits (0 to 12) or equivalent |
| Displacement is 61.2m | A1 | 5 |
(i) $a(t) = 1.25 - 0.1t$ | B1 | May be scored in (ii)
Initial acceleration is 1.25$ms^{-2}$ | B1 | 2 | Must follow an attempt to differentiate
(ii) $[t = 12]$ | M1 | For attempting to solve $dv/dt = 0.05$
$1.25t^2/2 - 0.05t^3$ $(+C)$ | M1, A1 | For attempting to integrate $v(t)$
$[1.25 \times 12^2/2 - 0.05 \times 12^3/3 = 90 - 28.8]$ | DM1 | For using appropriate limits (0 to 12) or equivalent
Displacement is 61.2m | A1 | 54 The velocity of a particle $t \mathrm {~s}$ after it starts from rest is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $v = 1.25 t - 0.05 t ^ { 2 }$. Find\\
(i) the initial acceleration of the particle,\\
(ii) the displacement of the particle from its starting point at the instant when its acceleration is $0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
\hfill \mbox{\textit{CAIE M1 2006 Q4 [7]}}