A particle \(P\) of mass 0.5 kg falls vertically from rest. After \(t\) seconds it has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude 1.5 v newtons acts on \(P\) as it falls.
Show that \(3 v = 9.8 \left( 1 - \mathrm { e } ^ { - 3 t } \right)\).
Find the distance that \(P\) falls in the first two seconds of its motion.
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A river is 50 m wide and flows between two straight parallel banks. The river flows with a uniform speed of \(\frac { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to the banks. The points \(A\) and \(B\) are on opposite banks of the river and \(A B\) is perpendicular to both banks of the river, as shown in Figure 1.
Keith and Ian decide to swim across the river. The speed relative to the water of both swimmers is \(\frac { 10 } { 9 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Keith sets out from \(A\) and crosses the river in the least possible time, reaching the opposite bank at the point \(C\). Find
the time taken by Keith to reach \(C\),
the distance \(B C\).
Ian sets out from \(A\) and swims in a straight line so as to land on the opposite bank at \(B\).