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

Using the equation of motion along the plane:
$$mv\frac{dv}{dx}=mg\sin\alpha-\frac13mx^2$$$$\frac12v^2=xg\sin\alpha-\frac19x^3$$

At x=2:
$$\frac12v^2=2g\sin\alpha-\frac89$$

Since $\sin\alpha=\frac25$,
$$v=3.7\ \text{or}\ 3.73\text{ m s}^{-1}$$

M1 A1 DM1 A1

Alternative energy method:
$$mg\sin\alpha\,x=\int\frac13mx^2\,dx+\frac12mv^2$$

Marking guidance for this part:
M1 for attempting an equation of motion parallel to the plane.
A1 for correct integration.
DM1 for substituting x=2 into the expression for $v^2$.
A1 for 3.7 or $3.73\text{ m s}^{-1}$.

$$v=0\Rightarrow x^2=9g\sin\alpha=9g\times\frac25,\qquad x\ne0$$$$x=5.939\ldots$$$$OA=5.9\text{ or }5.94\text{ m}$$
Marking guidance for this part:
M1 for using v=0 in their expression for v and obtaining a value of x.
A1 for 5.9 or $5.94\text{ m}$.