a) $0.24g = 12\left( x \right)/0.5$

$x = 0.1$

EITHER

$\frac{1}{2} \times {\text{0}}.24 \times {3^2} + 12 \times {\left( {0.8 - 0.5} \right)^2}/\left( {2 \times 0.5} \right) = $

$0.24{v^2}/2 + 12 \times {0.1^2}/\left( {2 \times 0.5} \right)$

$ + 0.24g\left( {0.8 - 0.5 - 0.1} \right)$

$v = 3.61{\text{ }}m{s^{ - 1}}$

OR

$0.24vdv/dx = mg - 12x/0.5$

$0.24{v^2}/2 = 2.4x - 12{x^2}{\text{ }}\left( { + c} \right)$

$v = 3,{\text{ }}x = 0.3,{\text{ }}c = 1.44$

$x = 0.1,{\text{ }}v = 3.61{\text{ }}m{s^{ - 1}}$

b) $0.24 \times {3^2}/2 + 12 \times {\left( {0.8 - 0.5} \right)^2}/\left( {2 \times 0.5} \right) = $

$0.24g\left( {0.8 + x} \right)$

$x = 0.1m$

$s = \left( {0.5 + 0.1} \right) = 0.6{\text{ }}m$

OR

$\frac{1}{2} \times 12 \times {0.3^2}/0.5 + \frac{1}{2} \times 0.24 \times {3^2}$

$ = \frac{1}{2} \times 0.24{v^2} + 0.24 \times 10 \times 0.3$

$v = \sqrt {12} $

Either

$0 = 12 - 2 \times 10s$

$s = 0.6$

OR $\frac{1}{2} \times 0.24 \times 12 = 0.24 \times 10s$

$s = 0.6$

OR

$\frac{1}{2} \times 12 \times {0.3^2}/0.5 + \frac{1}{2} \times 0.24 \times {3^2}$

$ = 0.24 \times 10y$

$y = 0.9$

$s = 0.9 - 0.3 = 0.6$