| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2008 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Maximum or minimum velocity |
| Difficulty | Moderate -0.3 This is a straightforward mechanics question requiring standard techniques: finding maximum velocity by differentiation (dv/dt = 0) and calculating distance as area under v-t graph using trapezium rule and integration. While it involves multiple steps and integration of a quadratic, these are routine M1 skills with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.08d Evaluate definite integrals: between limits3.02b Kinematic graphs: displacement-time and velocity-time3.02c Interpret kinematic graphs: gradient and area |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((dv/dt) = -0.02t + 0.5\) or \(v = -0.01[(t-T)^2 - 100V]\) where \(T = 25\) and \(V = 5.25\) (or equivalent) | B1 | |
| M1 | For solving \(dv/dt = 0\) or for selecting \(t = T\) or \(v_{max} = V\); may be implied when \(v_{max} = V\) is selected and T is 25 in the 'B1' expression for \(v\) | |
| \(t = 25\) | A1 | |
| Maximum velocity is \(5.25 \text{ ms}^{-1}\) | A1 [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(s_2 = -0.01t^3/3 + 0.5t^2/2 - t\) | M1, A1 | For integrating \(v(t)\) |
| M1 | For using limits 10 and 30 | |
| \(s_2 = (-90 + 225 - 30) - (-10/3 + 25 - 10)\ (= 93.3 \text{ m})\) | A1 | |
| M1 | For evaluating \(v(10)\) and \(v(30)\) | |
| \(v(10) = 3\) and \(v(30) = 5\) | A1 | |
| M1 | For evaluating \(s_1\) and \(s_3\) | |
| \(s_1 = \frac{1}{2}\times3\times10\) and \(s_3 = \frac{1}{2}\times5\times50\) | A1ft | ft incorrect values of \(v(10)\) and/or \(v(30)\) |
| Distance is 233 m | A1ft [9] | ft \(140 + s_2\) (depends on 1st M1) |
| SR for candidates treating first line segment as part of curve in part (ii) (max. mark 6/9): Integration M1 A1 as scheme; \(s_1 + s_2 = 105\) — A1; \(v(30) = 5\) — B1; \(s_3 = \frac{1}{2}\times5\times50\) — B1ft; Distance is 230 m — A1ft (ft \(125 + s_1 + s_2\)) |
## Question 7:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(dv/dt) = -0.02t + 0.5$ **or** $v = -0.01[(t-T)^2 - 100V]$ where $T = 25$ and $V = 5.25$ (or equivalent) | B1 | |
| | M1 | For solving $dv/dt = 0$ or for selecting $t = T$ or $v_{max} = V$; may be implied when $v_{max} = V$ is selected and T is 25 in the 'B1' expression for $v$ |
| $t = 25$ | A1 | |
| Maximum velocity is $5.25 \text{ ms}^{-1}$ | A1 [4] | |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $s_2 = -0.01t^3/3 + 0.5t^2/2 - t$ | M1, A1 | For integrating $v(t)$ |
| | M1 | For using limits 10 and 30 |
| $s_2 = (-90 + 225 - 30) - (-10/3 + 25 - 10)\ (= 93.3 \text{ m})$ | A1 | |
| | M1 | For evaluating $v(10)$ and $v(30)$ |
| $v(10) = 3$ and $v(30) = 5$ | A1 | |
| | M1 | For evaluating $s_1$ and $s_3$ |
| $s_1 = \frac{1}{2}\times3\times10$ and $s_3 = \frac{1}{2}\times5\times50$ | A1ft | ft incorrect values of $v(10)$ and/or $v(30)$ |
| Distance is 233 m | A1ft [9] | ft $140 + s_2$ (depends on 1st M1) |
| **SR** for candidates treating first line segment as part of curve in part **(ii)** (max. mark 6/9): Integration M1 A1 as scheme; $s_1 + s_2 = 105$ — A1; $v(30) = 5$ — B1; $s_3 = \frac{1}{2}\times5\times50$ — B1ft; Distance is 230 m — A1ft (ft $125 + s_1 + s_2$) | | |7\\
\includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-4_719_1059_264_543}
An object $P$ travels from $A$ to $B$ in a time of 80 s . The diagram shows the graph of $v$ against $t$, where $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ is the velocity of $P$ at time $t \mathrm {~s}$ after leaving $A$. The graph consists of straight line segments for the intervals $0 \leqslant t \leqslant 10$ and $30 \leqslant t \leqslant 80$, and a curved section whose equation is $v = - 0.01 t ^ { 2 } + 0.5 t - 1$ for $10 \leqslant t \leqslant 30$. Find\\
(i) the maximum velocity of $P$,\\
(ii) the distance $A B$.
\hfill \mbox{\textit{CAIE M1 2008 Q7 [13]}}