Distance traveled with variable force

4 A particle \(P\) of mass 0.2 kg travels in a straight line on a horizontal surface. It passes through a point \(O\) on the surface with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resistive force of magnitude \(0.2 \left( v + v ^ { 2 } \right) \mathrm { N }\) acts on \(P\) in the direction opposite to its motion, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\) when it is at a distance \(x \mathrm {~m}\) from \(O\).

1. Show that \(\frac { 1 } { 1 + v } \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 1\).
2. By solving the differential equation in part (i) show that \(\frac { - \mathrm { e } ^ { x } } { 3 - \mathrm { e } ^ { x } } \frac { \mathrm {~d} x } { \mathrm {~d} t } = - 1\), where \(t\) s is the time taken for \(P\) to travel \(x \mathrm {~m}\) from \(O\).
3. Hence find the value of \(t\) when \(x = 1\).

2 A duck of mass 2 kg is travelling with horizontal speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it lands on a lake. The duck is brought to rest by the action of resistive forces, acting in the direction opposite to the duck's motion and having total magnitude \(\left( 2 v + 3 v ^ { 2 } \right) \mathrm { N }\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the duck. Show that the duck comes to rest after travelling approximately 1.30 m from the point of its initial contact with the surface of the lake.

5. A particle \(Q\) of mass 6 kg is moving along the \(x\)-axis. At time \(t\) seconds the displacement of \(Q\) from the origin \(O\) is \(x\) metres and the speed of \(Q\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle moves under the action of a retarding force of magnitude ( \(a + b v ^ { 2 }\) ) N, where \(a\) and \(b\) are positive constants. At time \(t = 0 , Q\) is at \(O\) and moving with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. The particle \(Q\) comes to instantaneous rest at the point \(X\).

1. Show that the distance \(O X\) is $$\frac { 3 } { b } \ln \left( 1 + \frac { b U ^ { 2 } } { a } \right) \mathrm { m }$$ Given that \(a = 12\) and \(b = 3\),
2. find, in terms of \(U\), the time taken to move from \(O\) to \(X\).

7 A small ball \(B\) of mass 0.2 kg moves in a narrow fixed smooth cylindrical tube \(O A\) of length 1 m , closed at the end \(A\). When the ball has displacement \(x \mathrm {~m}\) from \(O\), it has velocity \(v \mathrm {~ms} ^ { - 1 }\) in the direction \(O A\) and experiences a resisting force of magnitude \(\frac { k } { 1 - x } \mathrm {~N}\).

1. \includegraphics[max width=\textwidth, alt={}, center]{10abedc3-c814-47c0-8ed4-849ef325feca-4_186_805_488_715} The tube is fixed in a horizontal position and \(B\) is projected from \(O\) towards \(A\) with velocity \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Given that \(B\) comes to instantaneous rest after travelling 0.55 m , show that \(k = 0.1803\), correct to 4 significant figures.
2. The tube is now fixed in a vertical position with \(O\) above \(A\). The ball \(B\) is released from rest at \(O\). Calculate the speed of \(B\) after it has descended 0.1 m . \end{document}