Standard +0.8 This is a variable force problem requiring the v dv/dx method with integration. Students must set up F = ma as 0.3x = 0.6v dv/dx, separate variables, and integrate to find v when x = 8. While the integration itself is straightforward, recognizing the appropriate method and correctly applying it with initial conditions requires solid understanding beyond routine exercises. This is moderately challenging for M2 level.
1 A particle \(P\) of mass 0.6 kg is projected horizontally with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on a smooth horizontal surface. A horizontal force of magnitude \(0.3 x \mathrm {~N}\) acts on \(P\) in the direction \(O P\), where \(x \mathrm {~m}\) is the distance of \(P\) from \(O\). Calculate the velocity of \(P\) when \(x = 8\).
1 A particle $P$ of mass 0.6 kg is projected horizontally with velocity $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point $O$ on a smooth horizontal surface. A horizontal force of magnitude $0.3 x \mathrm {~N}$ acts on $P$ in the direction $O P$, where $x \mathrm {~m}$ is the distance of $P$ from $O$. Calculate the velocity of $P$ when $x = 8$.
\hfill \mbox{\textit{CAIE M2 2012 Q1 [4]}}