Moderate -0.5 This is a straightforward calculus application requiring differentiation of a polynomial to find velocity, setting v=0, and substituting back to find displacement. It's slightly easier than average as it involves only basic differentiation and solving a quadratic equation with no conceptual complications.
3 A particle moves along a straight line containing a point O . Its displacement, \(x \mathrm {~m}\), from O at time \(t\) seconds is given by
$$x = 12 t - t ^ { 3 } , \text { where } - 10 \leqslant t \leqslant 10$$
Find the values of \(x\) for which the velocity of the particle is zero.
Accept one answer only but no extra answers. FT only if quadratic or higher degree
\(x = \pm 16\)
A1
cao. Must have both and no extra answers
Total: 5 marks
## Question 3:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(v =) 12 - 3t^2$ | M1 | Differentiating |
| $(v =) 12 - 3t^2$ (correct) | A1 | Allow confusion of notation, including $x =$ |
| $v = 0 \Rightarrow 12 - 3t^2 = 0$ | M1 | Dep on 1st M1. Equating to zero |
| so $t^2 = 4$ and $t = \pm 2$ | A1 | Accept one answer only but no extra answers. FT only if quadratic or higher degree |
| $x = \pm 16$ | A1 | cao. Must have both and no extra answers |
**Total: 5 marks**
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3 A particle moves along a straight line containing a point O . Its displacement, $x \mathrm {~m}$, from O at time $t$ seconds is given by
$$x = 12 t - t ^ { 3 } , \text { where } - 10 \leqslant t \leqslant 10$$
Find the values of $x$ for which the velocity of the particle is zero.
\hfill \mbox{\textit{OCR MEI M1 Q3 [5]}}