A particle \(P\) moves along the \(x\)-axis. At time \(t = 0 , P\) passes through the origin \(O\), moving in the positive \(x\)-direction. At time \(t\) seconds, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(O P = x\) metres. The acceleration of \(P\) is \(\frac { 1 } { 12 } ( 30 - x ) \mathrm { m } \mathrm { s } ^ { - 2 }\), measured in the positive \(x\)-direction.
Give a reason why the maximum speed of \(P\) occurs when \(x = 30\).
Given that the maximum speed of \(P\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
find an expression for \(v ^ { 2 }\) in terms of \(x\).