Edexcel IAL Mathematics M3.1.1 straight-line motion with variable acceleration

Practise straight-line particle kinematics where acceleration depends on x, v or t, using differentiation, integration and initial conditions.

[Maximum number: 7]

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

$t=\frac{a+\sqrt{b x^{2}+c}}{d}$ , where a, b, c and d are integers to be found.

$\frac{\mathrm{d} x}{\mathrm{~d} t}=\sqrt{\frac{4}{x^{2}}+5}$

M1

$\int \frac{x}{\sqrt{5 x^{2}+4}} \mathrm{~d} x=\int \mathrm{d} t$

M1A1

$\frac{1}{5} \sqrt{5 x^{2}+4}=t \quad\left(+k_{1}\right)$

DM1A1

$x=1, t=1 \Rightarrow k_{1}=-\frac{2}{5}$

DM1

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

A1

(7)
(a)

M1 Dimensionally correct equation of motion. Acceleration must be in form $v \frac{\mathrm{~d} v}{\mathrm{~d} x}$ . Condone missing minus sign.

DM1 Separate variables and attempt integration. Condone missing minus sign.
A1 Correct integrals. Condone missing constant.
DM1 Use v=3, x=1 to find constant of integration. Dependent on both previous M marks.
A1* Reach given result with no errors.

Alt - Definite Integration/Energy Work.
M1 - Equate change in KE to Integral for WD. Integration not needed for this mark and condone inconsistent signs.

\frac{1}{2} \times 2 v^{2}-\frac{1}{2} \times 2 \times 3^{2}=\int( \pm) \frac{2}{x^{3}} d x

DM1 Attempt integration. Condone inconsistent signs.
A1 Correct integration, with consistent signs for KE and WD.
DM1 Substitute in limits.
A1 Reach given result with no errors.
(b)

M1 Use $v=\frac{\mathrm{d} x}{\mathrm{~d} t}$ to form differential equation in x and t.
M1 Separate variables and to produce functions ready to be integrated.
A1 Correct integrands, written in a form that can be integrated.
DM1 Valid attempt to integrate their expression. If they have an incorrect expression, the integration must not be significantly simplified. Dependent on first 2 M marks.

A1 Correct integration. Condone missing constant.
DM1 Use x=1, t=1 to find constant of integration. Dependent on all 3 M marks.
A1 Correct result. (cso)

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M3.1.1 - Kinematics of a particle moving question 1