Edexcel A-Level Mathematics A2 M3.1.1 Kinematics of a Particle Moving Questions

Practise straight-line variable-acceleration problems by converting between a, v, x and t, using differentiation or integration, and applying given initial positions and times.

Exam points

convert between a, v, x and t using a = v dv/dx, dv/dt and v = dx/dt
integrate a velocity or acceleration relation and use given x, t values to find constants

[Maximum number: 10]

In this question you must show all stages in your working.

Solutions relying entirely on calculator technology are not acceptable.

A particle P is moving along the x-axis.
At time t seconds, where $0 \leqslant t \leqslant \frac{2}{3}, P$ is x metres from the origin O and is moving with velocity $v \mathrm{~m} \mathrm{~s}^{-1}$ in the positive x direction where

v=(2 x+1)^{\frac{3}{2}}

When t=0, P passes through O.

(a)

Find the value of x when the acceleration of P is $243 \mathrm{~m} \mathrm{~s}^{-2}$

[ 4 ]

$a=v \frac{\mathrm{~d} v}{\mathrm{~d} x}$ $=\frac{3}{2}(2 x+1)^{\frac{1}{2}} \times 2 \times(2 x+1)^{\frac{3}{2}}=3(2 x+1)^{2}$ $3(2 x+1)^{2}=243$
x=4
(4)

(b)

Find v in terms of t.

[ 6 ]

$(2x+1)^{3/2}=\frac{\mathrm{d}x}{\mathrm{d}t}$
or $a=3v^{4/3}=\frac{\mathrm{d}v}{\mathrm{d}t}$ $\mathrm{d}t=(2x+1)^{-3/2}\,\mathrm{d}x$ $t=-(2x+1)^{-1/2}+C$ $t=0,\ x=0\Rightarrow C=1$
and obtain an equation in v and t only.
$v=\frac{1}{(1-t)^3}$

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Edexcel A-Level Mathematics A2 M3.1.1 Kinematics of a Particle Moving Questions question 1

Question (a)

Question (b)