8 A piston of mass 1.5 kg moves in a straight line inside a long straight horizontal cylinder. At time \(t \mathrm {~s}\) the displacement of the piston from its initial position at one end of the cylinder is \(x \mathrm {~m}\) and its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{a8c9d007-e67f-4637-9e74-630ba9a91442-5_168_805_1726_630} The piston starts moving when \(t = 2\) and is brought to rest when it reaches the other end of the cylinder. While the piston is in motion it is acted on by a force of magnitude \(\frac { 6 } { t ^ { 2 } } \mathrm {~N}\) in the positive \(x\) direction, and also by a force of magnitude \(\frac { 3 v } { t } \mathrm {~N}\) resisting the motion.

1. Show that, while the piston is in motion, \(\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 2 v } { t } = \frac { 4 } { t ^ { 2 } }\). The piston reaches the other end of the cylinder when \(t = 20\).
2. Find the speed of the piston immediately before it is brought to rest.
3. Show that the piston travels a distance of 5.61 m , correct to 3 significant figures. \section*{OCR} \section*{Oxford Cambridge and RSA}