| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2007 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Piecewise motion functions |
| Difficulty | Standard +0.3 This is a standard M2 piecewise velocity question requiring routine calculus techniques: differentiation to find maximum speed, integration for displacement, solving v=0 for rest time, and careful tracking of direction changes for total distance. While multi-part with several steps, each component uses straightforward methods without requiring novel insight or complex problem-solving. |
| Spec | 3.02a Kinematics language: position, displacement, velocity, acceleration3.02c Interpret kinematic graphs: gradient and area3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(0 \leq t \leq 4\): \(a = 8 - 3t\) | M1 | Differentiate to obtain acceleration |
| \(a = 0 \Rightarrow t = \frac{8}{3}\) s | DM1 | Set acceleration \(= 0\) and solve for \(t\) |
| \(v = 8 \cdot \frac{8}{3} - \frac{3}{2}\left(\frac{8}{3}\right)^2 = \frac{32}{3}\) m/s | DM1 A1 | Use their \(t\) to find \(v\); \(\frac{32}{3}\) or 10.7 or better |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(s = 4t^2 - \frac{t^3}{2}\) | M1 | Integrate the correct expression |
| \(t = 4\): \(s = 64 - \frac{64}{2} = 32\) m | M1 A1 | Substitute \(t=4\); condone omission/ignoring constant of integration; 32(m) only |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(t > 4\): \(v = 0 \Rightarrow t = 8\) s | B1 | \(t = 8\) (s) only |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(t > 4\): \(s = 16t - t^2 \ (+C)\) | M1 | Integrate \(16 - 2t\) |
| \(t = 4, s = 32 \Rightarrow C = -16 \Rightarrow s = 16t - t^2 - 16\) | M1 A1 | Use \(t=4\), \(s=\) their value from (b) to find constant; \(s = 16t - t^2 - 16\) or equivalent |
| \(t = 10 \rightarrow s = 44\) m | M1 A1 | Substitute \(t = 10\); 44 |
| Direction changed, so: \(t = 8\), \(s = 48\) | M1 | Substitute \(t = 8\) (their value from (c)) |
| Total distance \(= 48 + 4 = 52\) m | DM1 A1 | Calculate total distance; 52 (m) |
# Question 8:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $0 \leq t \leq 4$: $a = 8 - 3t$ | M1 | Differentiate to obtain acceleration |
| $a = 0 \Rightarrow t = \frac{8}{3}$ s | DM1 | Set acceleration $= 0$ and solve for $t$ |
| $v = 8 \cdot \frac{8}{3} - \frac{3}{2}\left(\frac{8}{3}\right)^2 = \frac{32}{3}$ m/s | DM1 A1 | Use their $t$ to find $v$; $\frac{32}{3}$ or 10.7 or better |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $s = 4t^2 - \frac{t^3}{2}$ | M1 | Integrate the correct expression |
| $t = 4$: $s = 64 - \frac{64}{2} = 32$ m | M1 A1 | Substitute $t=4$; condone omission/ignoring constant of integration; 32(m) only |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $t > 4$: $v = 0 \Rightarrow t = 8$ s | B1 | $t = 8$ (s) only |
## Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $t > 4$: $s = 16t - t^2 \ (+C)$ | M1 | Integrate $16 - 2t$ |
| $t = 4, s = 32 \Rightarrow C = -16 \Rightarrow s = 16t - t^2 - 16$ | M1 A1 | Use $t=4$, $s=$ their value from (b) to find constant; $s = 16t - t^2 - 16$ or equivalent |
| $t = 10 \rightarrow s = 44$ m | M1 A1 | Substitute $t = 10$; 44 |
| Direction changed, so: $t = 8$, $s = 48$ | M1 | Substitute $t = 8$ (their value from (c)) |
| Total distance $= 48 + 4 = 52$ m | DM1 A1 | Calculate total distance; 52 (m) |\begin{enumerate}
\item A particle $P$ moves on the $x$-axis. At time $t$ seconds the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ in the direction of $x$ increasing, where $v$ is given by
\end{enumerate}
$$v = \left\{ \begin{array} { l c }
8 t - \frac { 3 } { 2 } t ^ { 2 } , & 0 \leqslant t \leqslant 4 , \\
16 - 2 t , & t > 4 .
\end{array} \right.$$
When $t = 0 , P$ is at the origin $O$.\\
Find\\
(a) the greatest speed of $P$ in the interval $0 \leqslant t \leqslant 4$,\\
(b) the distance of $P$ from $O$ when $t = 4$,\\
(c) the time at which $P$ is instantaneously at rest for $t > 4$,\\
(d) the total distance travelled by $P$ in the first 10 s of its motion.\\
\hfill \mbox{\textit{Edexcel M2 2007 Q8 [16]}}