| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2011 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Maximum or minimum velocity |
| Difficulty | Standard +0.3 This is a straightforward non-constant acceleration question requiring standard calculus techniques: solving v=0 for part (i), finding dv/dt=0 for maximum speed in (ii), integrating velocity for distance in (iii), and solving s=0 for part (iv). All steps are routine M1 procedures with no conceptual challenges, though the fractional power and four-part structure place it slightly above average difficulty. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(v(100) = 0.16 \times 1000 - 0.016 \times 10000 = 0\) | B1 | 1 AG |
| (ii) \(a = 1.5 \times 0.16t^{-\frac{1}{2}} - 0.032t\) | M1 | For using \(a = dv/dt\) |
| A1 | ||
| \([t^{\frac{3}{2}} = 0.24/0.032 \Rightarrow t = 56.25 \Rightarrow v_{\max} = 0.16 \times 421.875 - 0.016 \times 3164.0625]\) | M1 | For solving \(a = 0\) and subst into \(v(t)\) |
| Maximum speed is 16.9 ms\(^{-1}\) (or \(16\frac{7}{8}\) ms\(^{-1}\)) | A1 | 4 |
| (iii) \(s = 2/5 \times 0.16t^{\frac{3}{2}} - 0.016t^3/3\) | M1 | For using \(s = \int v\,dt\) |
| A1 | ||
| Distance is 1070 m | A1 | 3 |
| (iv) \(\frac{1}{3}t^{\frac{3}{2}}(0.192 - 0.016\sqrt{t}) = 0\) | M1 | For attempting to solve \(s(t) = 0\) |
| Value of \(t\) is 144 | A1 | 2 |
| **(i)** $v(100) = 0.16 \times 1000 - 0.016 \times 10000 = 0$ | B1 | 1 AG |
| **(ii)** $a = 1.5 \times 0.16t^{-\frac{1}{2}} - 0.032t$ | M1 | For using $a = dv/dt$ |
| | A1 | |
| $[t^{\frac{3}{2}} = 0.24/0.032 \Rightarrow t = 56.25 \Rightarrow v_{\max} = 0.16 \times 421.875 - 0.016 \times 3164.0625]$ | M1 | For solving $a = 0$ and subst into $v(t)$ |
| Maximum speed is 16.9 ms$^{-1}$ (or $16\frac{7}{8}$ ms$^{-1}$) | A1 | 4 |
| **(iii)** $s = 2/5 \times 0.16t^{\frac{3}{2}} - 0.016t^3/3$ | M1 | For using $s = \int v\,dt$ |
| | A1 | |
| Distance is 1070 m | A1 | 3 |
| **(iv)** $\frac{1}{3}t^{\frac{3}{2}}(0.192 - 0.016\sqrt{t}) = 0$ | M1 | For attempting to solve $s(t) = 0$ |
| Value of $t$ is 144 | A1 | 2 |7 A particle $P$ starts from a point $O$ and moves along a straight line. $P$ 's velocity $t$ s after leaving $O$ is $\nu \mathrm { m } \mathrm { s } ^ { - 1 }$, where
$$v = 0.16 t ^ { \frac { 3 } { 2 } } - 0.016 t ^ { 2 } .$$
$P$ comes to rest instantaneously at the point $A$.\\
(i) Verify that the value of $t$ when $P$ is at $A$ is 100 .\\
(ii) Find the maximum speed of $P$ in the interval $0 < t < 100$.\\
(iii) Find the distance $O A$.\\
(iv) Find the value of $t$ when $P$ passes through $O$ on returning from $A$.
\hfill \mbox{\textit{CAIE M1 2011 Q7 [10]}}