| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Topic | Variable acceleration (1D) |
| Type | Maximum or minimum velocity |
| Difficulty | Standard +0.3 This is a straightforward variable acceleration question requiring standard calculus techniques: (i) integrate velocity to find displacement over an interval, (ii) differentiate velocity to find acceleration, set to zero, and verify maximum using second derivative test. Both parts are routine A-level mechanics applications with no conceptual challenges beyond textbook exercises. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives3.02f Non-uniform acceleration: using differentiation and integration |
**Question 10(i)**
Attempt to integrate $v$ **M1**
Use correct limits **M1** dep on first **M1**; Allow $\pm[f(4) - f(1)]$
$18.3\ \text{m}$ **A1** (18.28125)
**Question 10(ii)**
Attempt to differentiate $v$ **M1**
Equate differential to $0$ and solve to get $t = 2\ \text{s}$ **A1**
Test for a maximum **B1** Show a sketch, at least $t = 2$ marked.10 A particle $P$ moves in a straight line starting from $O$. At time $t$ seconds after leaving $O$, the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $v = 5 + 1.5 t - 0.125 t ^ { 3 }$.\\
(i) Find the displacement of $P$ between the times $t = 1$ and $t = 4$.\\
(ii) Find the time at which the velocity of $P$ is a maximum, justifying your answer.
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2018 Q10 [7]}}