Standard +0.8 This M4 question requires setting up Newton's second law with v²-resistance and buoyancy, then using the chain rule (v dv/dx = a) to derive a differential equation. Part (b) requires integration and finding where v=0. While methodical, it demands careful force analysis, correct application of calculus techniques for variable force, and numerical solution—more challenging than standard mechanics but follows established M4 patterns.
A wooden ball of mass 0.01 kg falls vertically into a pond of water. The speed of the ball as it enters the water is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the ball is \(x\) metres below the surface of the water and moving downwards with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the water provides a resistance of magnitude \(0.02 v ^ { 2 } \mathrm {~N}\) and an upward buoyancy force of magnitude 0.158 N .
Show that, while the ball is moving downwards,
$$- 2 v ^ { 2 } - 6 = v \frac { \mathrm {~d} v } { \mathrm {~d} x }$$
Hence find, to 3 significant figures, the greatest distance below the surface of the water reached by the ball.
\begin{enumerate}
\item A wooden ball of mass 0.01 kg falls vertically into a pond of water. The speed of the ball as it enters the water is $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When the ball is $x$ metres below the surface of the water and moving downwards with speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$, the water provides a resistance of magnitude $0.02 v ^ { 2 } \mathrm {~N}$ and an upward buoyancy force of magnitude 0.158 N .\\
(a) Show that, while the ball is moving downwards,
\end{enumerate}
$$- 2 v ^ { 2 } - 6 = v \frac { \mathrm {~d} v } { \mathrm {~d} x }$$
(b) Hence find, to 3 significant figures, the greatest distance below the surface of the water reached by the ball.\\
\hfill \mbox{\textit{Edexcel M4 2003 Q1 [8]}}