| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| Topic | Travel graphs |
| Type | Model refinement or criticism |
| Difficulty | Moderate -0.8 This is a straightforward kinematics question requiring only standard SUVAT equations and basic calculus (integration to find displacement). Part (i) uses constant acceleration formulas, part (ii)(a) is trivial substitution, part (ii)(b) requires integrating the given velocity function, and part (iii) asks for a qualitative comparison. All techniques are routine A-level mechanics with no problem-solving insight required, making it easier than average. |
| Spec | 3.02d Constant acceleration: SUVAT formulae3.02f Non-uniform acceleration: using differentiation and integration |
**Question 9(i)**
- $3.6 = \dfrac{1}{2}(0 + v) \times 18$ **M1** (Use of appropriate 'suvat' formula.)
- $\therefore v = 0.4$ (ms$^{-1}$) **A1** (cao)
**Question 9(ii)(a)**
- $t = 0$ or $18$. **B1** (Both values required.)
**Question 9(ii)(b)**
- Integrate $v$. **M1**
- $x = \displaystyle\int \dfrac{1}{270}(18t - t^2)\,\mathrm{d}t = \dfrac{1}{270}\left(9t^2 - \dfrac{t^3}{3}\right) + c$ **A1** (All terms correct; condone omission of "$+ c$". Allow definite integral as alternative.)
- When $t = 0$, $x = 2$, $\therefore c = 2$ **M1** (Deal with $c$ correctly or apply limits of definite integral.)
- When $t = 18$: $I = \dfrac{1}{270}\left(9 \times 18^2 - \dfrac{18^3}{3}\right) = \dfrac{2916 - 1944}{270} = \dfrac{972}{270} = 3.6$
- $\therefore x = 3.6 + 2 = 5.6$ **A1** (Evaluate for $t = 18$ or add 2 if definite integral used. Convincingly shown.)
**Question 9(iii)**
- In Model 2, $v = 0$ when the particle reaches $Q$. **B1**
**Total: 8 marks**9 A particle moves along a straight line such that its displacement from $O$, a fixed point on the line, is $x$. The particle travels from rest from the point $P$, where $x = 2$, to the point $Q$, where $x = 5.6$. All distances are in metres. Two models for the motion of the particle are proposed.\\
(i) In Model 1, the acceleration of the particle is assumed to be constant and the particle takes 18 seconds to travel from $P$ to $Q$. Find the velocity of the particle when it reaches $Q$.\\
(ii) In Model 2, the velocity after $t$ seconds is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, where $v = \frac { 1 } { 270 } \left( 18 t - t ^ { 2 } \right)$.
\begin{enumerate}[label=(\alph*)]
\item Write down the values of $t$ when $v = 0$.
\item Show that $x = 5.6$ when $t = 18$.\\
(iii) The particle represents a fragile instrument that is being moved from $P$ to $Q$ across a laboratory. Explain why Model 2 might be more appropriate than Model 1.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2017 Q9 [8]}}