Edexcel A-Level Mathematics A2 M3.3.1 Variable Force Motion in One Dimension Questions

Practise one-dimensional motion with variable forces, using Newton's laws, calculus and energy methods to find speeds and stopping distances.

Exam points

convert a variable-force equation into v dv/dx and integrate to obtain v^2
use initial speed or position data to determine the integration constant
find a speed or stopping distance by substituting a stated value of x

[Maximum number: 5]

A particle P of mass 0.5 kg moves along the positive x-axis in the positive x direction.

At time t seconds, $t \geqslant 1, P$ is x metres from the origin O and is moving with speed $v \mathrm{~m} \mathrm{~s}^{-1}$ . The resultant force acting on P has magnitude $\frac{2}{x^{3}} \mathrm{~N}$ and is directed towards O.

When t=1, x=1 and v=3

Show that

$v^{2}=\frac{4}{x^{2}}+5$

$0.5 v \frac{\mathrm{~d} v}{\mathrm{~d} x}=-\frac{2}{x^{3}}$ $\int v \mathrm{~d} v=\int-\frac{4}{x^{3}} \mathrm{~d} x$ $\frac{v^{2}}{2}=\frac{2}{x^{2}} \quad(+k)$ $v^{2}=\frac{4}{x^{2}}+2 k$ $v=3, x=1 \Rightarrow 2 k=5$ $v^{2}=\frac{4}{x^{2}}+5 *$

A1 *

(5)

Alt for first 2 marks using Energy

Use of $W=\int F d x$
Attempt integration and equate to change in KE.

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Edexcel A-Level Mathematics A2 M3.3.1 Variable Force Motion in One Dimension Questions question 1