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

$\frac{\mathrm{d} x}{\mathrm{~d} t}=\sqrt{\frac{4}{x^{2}}+5}$$\int \frac{x}{\sqrt{5 x^{2}+4}} \mathrm{~d} x=\int \mathrm{d} t$$\frac{1}{5} \sqrt{5 x^{2}+4}=t \quad\left(+k_{1}\right)$$x=1, t=1 \Rightarrow k_{1}=-\frac{2}{5}$$t=\frac{2+\sqrt{5 x^{2}+4}}{5}$
(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.

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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