| Exam Board | OCR |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2010 |
| 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 M1 kinematics question requiring standard techniques: setting v=0 to find when at rest, differentiating v to find acceleration, and integrating v to find displacement. All steps are routine applications of calculus with no problem-solving insight needed, making it slightly easier than average. |
| Spec | 3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(3.2 - 0.2t^2 = 0\), \(t = 4\) s | M1, A1 | Puts 0 for v and attempts to solve QE. Accept dual solution +/-4 |
| Answer | Marks | Guidance |
|---|---|---|
| \(a = -2x0.2t\), \(a = -0.4x4\), \(a = -1.6\) m s\(^{-2}\) | M1*, D*M1, A1 | Differentiates v. Substitutes +ve t(i) in derivative of v. Negative only |
| Answer | Marks | Guidance |
|---|---|---|
| \(s = 3.2t - 0.2t^2/3\) (+c), \(t = 0, s = 0\) so \(c = 0\), \(s(4) = 3.2x4 - 0.2x4^3/3\), \(s = 8.53\) m | M1*, A1, B1, D*M1, A1 | Integrates v, not multiplication by t. Or correct use of limits 0 and 4. Accept without/loss of c. Accept with/without c |
**Part i**
| $3.2 - 0.2t^2 = 0$, $t = 4$ s | M1, A1 | Puts 0 for v and attempts to solve QE. Accept dual solution +/-4 |
|---|---|---|
**Part ii**
| $a = -2x0.2t$, $a = -0.4x4$, $a = -1.6$ m s$^{-2}$ | M1*, D*M1, A1 | Differentiates v. Substitutes +ve t(i) in derivative of v. Negative only |
**Part iii**
| $s = 3.2t - 0.2t^2/3$ (+c), $t = 0, s = 0$ so $c = 0$, $s(4) = 3.2x4 - 0.2x4^3/3$, $s = 8.53$ m | M1*, A1, B1, D*M1, A1 | Integrates v, not multiplication by t. Or correct use of limits 0 and 4. Accept without/loss of c. Accept with/without c |
---4 A particle $P$ moving in a straight line has velocity $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at time $t \mathrm {~s}$ after passing through a fixed point $O$. It is given that $v = 3.2 - 0.2 t ^ { 2 }$ for $0 \leqslant t \leqslant 5$. Calculate\\
(i) the value of $t$ when $P$ is at instantaneous rest,\\
(ii) the acceleration of $P$ when it is at instantaneous rest,\\
(iii) the greatest distance of $P$ from $O$.
\hfill \mbox{\textit{OCR M1 2010 Q4 [10]}}