9231 P33 - Jun 2025 - Q3 - 8 marks
6866
A ball of mass \(m\) kg is projected vertically upwards with initial speed \(U\,\text{m s}^{-1}\). At time \(t\), it has travelled distance \(x\) m and has speed \(v\,\text{m s}^{-1}\). There is a resistive force of magnitude \(mkv^2\) N, where \(k\gt0\).
(a) Show that, while the ball is moving upwards,
\(x=\frac{1}{2k}\ln\left(\frac{g+kU^2}{g+kv^2}\right).\)
(b) Given \(k=0.025\) and \(U=20\), find the time taken to reach maximum height.
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