Air resistance with other powers

4 A particle of mass 0.2 kg moves in a straight line on a smooth horizontal surface. When its displacement from a fixed point on the surface is \(x \mathrm {~m}\), its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The motion is opposed by a force of magnitude \(\frac { 1 } { 3 v } \mathrm {~N}\).

1. Show that \(3 v ^ { 2 } \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 5\).
2. Find the value of \(v\) when \(x = 7.4\), given that \(v = 4\) when \(x = 0\).

6 A particle \(P\) of mass 0.5 kg moves in a straight line on a smooth horizontal surface. At time \(t \mathrm {~s}\), the displacement of \(P\) from a fixed point on the line is \(x \mathrm {~m}\) and the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It is given that when \(t = 0 , x = 0\) and \(v = 9\). The motion of \(P\) is opposed by a force of magnitude \(3 \sqrt { } v \mathrm {~N}\).

1. By solving an appropriate differential equation, show that \(v = ( 27 - 9 x ) ^ { \frac { 2 } { 3 } }\).
2. Calculate the value of \(x\) when \(t = 0.5\).

5 A particle \(P\) of mass 0.4 kg moves in a straight line on a horizontal surface and has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\). A horizontal force of magnitude \(k \sqrt { } v \mathrm {~N}\) opposes the motion of \(P\). When \(t = 0 , v = 9\) and when \(t = 2 , v = 4\).

1. Express \(\frac { \mathrm { d } v } { \mathrm {~d} t }\) in terms of \(k\) and \(v\), and hence show that \(v = \frac { 1 } { 4 } ( t - 6 ) ^ { 2 }\).
2. Find the distance travelled by \(P\) in the first 3 seconds of its motion.

2 The resistance to the motion of a car is \(k v ^ { \frac { 3 } { 2 } } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the car's speed and \(k\) is a constant. The power exerted by the car's engine is 15000 W , and the car has constant speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a horizontal road.

1. Show that \(k = 4.8\). With the engine operating at a much lower power, the car descends a hill of inclination \(\alpha\), where \(\sin \alpha = \frac { 1 } { 15 }\). At an instant when the speed of the car is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Given that the mass of the car is 700 kg , calculate the power of the engine. \includegraphics[max width=\textwidth, alt={}, center]{941c0c81-a74f-49c0-acb7-1c23266fc2c8-02_579_447_1658_849} A particle \(P\) of mass 0.4 kg is attached to one end of each of two light inextensible strings which are both taut. The other end of the longer string is attached to a fixed point \(A\), and the other end of the shorter string is attached to a fixed point \(B\), which is vertically below \(A\). The string \(A P\) makes an angle of \(30 ^ { \circ }\) with the vertical and is 0.5 m long. The string \(B P\) makes an angle of \(60 ^ { \circ }\) with the vertical. \(P\) moves with constant angular speed in a horizontal circle with centre vertically below \(B\) (see diagram). The tension in the string \(A P\) is twice the tension in the string \(B P\). Calculate

5 A particle, of mass 12 kg , is moving along a straight horizontal line. At time \(t\) seconds, the particle has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As the particle moves, it experiences a resistance force of magnitude \(4 v ^ { \frac { 1 } { 3 } }\). No other horizontal force acts on the particle. The initial speed of the particle is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that $$v = \left( 4 - \frac { 2 } { 9 } t \right) ^ { \frac { 3 } { 2 } }$$
2. Find the value of \(t\) when the particle comes to rest.

7 A particle of mass 20 kg moves along a straight horizontal line. At time \(t\) seconds the velocity of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resistance force of magnitude \(10 \sqrt { v }\) newtons acts on the particle while it is moving. At time \(t = 0\) the velocity of the particle is \(25 \mathrm {~ms} ^ { - 1 }\).

1. Show that, at time \(t\) $$v = \left( \frac { 20 - t } { 4 } \right) ^ { 2 }$$
2. State the value of \(t\) when the particle comes to rest.

6 A car, of mass \(m \mathrm {~kg}\), is moving along a straight horizontal road. At time \(t\) seconds, the car has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). As the car moves, it experiences a resistance force of magnitude \(2 m v ^ { \frac { 5 } { 4 } }\) newtons. No other horizontal force acts on the car.

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 2 v ^ { \frac { 5 } { 4 } }$$ (1 mark)
2. The initial speed of the car is \(16 \mathrm {~ms} ^ { - 1 }\). Show that $$v = \left( \frac { 2 } { t + 1 } \right) ^ { 4 }$$ (5 marks)

3 A particle \(P\) of mass 0.3 kg is projected horizontally with speed \(u \mathrm {~ms} ^ { - 1 }\) from a fixed point \(O\) on a smooth horizontal surface. At time \(t \mathrm {~s}\) after projection \(P\) is \(x \mathrm {~m}\) from \(O\) and is moving with speed \(v \mathrm {~ms} ^ { - 1 }\). There is a force of magnitude \(1.2 v ^ { 3 } \mathrm {~N}\) resisting the motion of \(P\).

1. Find an expression for \(\frac { \mathrm { d } v } { \mathrm {~d} x }\) in terms of \(v\) and hence show that \(v = \frac { u } { 4 u x + 1 }\).
2. Given that \(x = 2\) when \(t = 9\) find the value of \(u\).

2 A particle of mass \(m \mathrm {~kg}\) moves horizontally in a straight line with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\). The total resistance force on the particle is of magnitude \(m k v ^ { \frac { 3 } { 2 } } \mathrm {~N}\) where \(k\) is a positive constant. There are no other horizontal forces present. Initially \(v = 25\) and the particle is at a point O .

1. Show that \(v = 4 \left( k t + \frac { 2 } { 5 } \right) ^ { - 2 }\).
2. Find the displacement from O of the particle at time \(t\).
3. Describe the motion of the particle as \(t\) increases. Section B (48 marks)

9 A particle of mass 2 kg is moving along the \(x\)-axis, which is horizontal, against a resistive force which is proportional to the cube of the speed of the particle at any instant. At time \(t\) seconds the particle's velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its displacement is \(x \mathrm {~m}\). When \(t = 0 , x = 0 , v = 4\) and the retardation is \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that $$\frac { 1 } { v } = \frac { x + 8 } { 32 } .$$
2. Find the time taken to cover the first 8 metres.

\(O\), \(A\) and \(B\) are three points in a straight line on a smooth horizontal surface. A particle \(P\) of mass \(0.6\) kg moves along the line. At time \(t\) s the particle has displacement \(x\) m from \(O\) and speed \(v\) m s\(^{-1}\). The only horizontal force acting on \(P\) has magnitude \(0.4v^{\frac{1}{2}}\) N and acts in the direction \(OA\). Initially the particle is at \(A\), where \(x = 1\) and \(v = 1\).

1. Show that \(3v^{\frac{1}{2}}\frac{dv}{dx} = 2\). [2]
2. Express \(v\) in terms of \(x\). [4]
3. Given that \(AB = 7\) m, find the value of \(t\) when \(P\) passes through \(B\). [3]

A puck, of mass \(m\) kg, is moving in a straight line across smooth horizontal ice. At time \(t\) seconds, the puck has speed \(v \text{ m s}^{-1}\). As the puck moves, it experiences an air resistance force of magnitude \(0.3mv^3\) newtons, until it comes to rest. No other horizontal forces act on the puck. When \(t = 0\), the speed of the puck is \(8 \text{ m s}^{-1}\). Model the puck as a particle.

1. Show that $$v = (4 - 0.2t)^{\frac{3}{2}}$$ [6 marks]
2. Find the value of \(t\) when the puck comes to rest. [2 marks]
3. Find the distance travelled by the puck as its speed decreases from \(8 \text{ m s}^{-1}\) to zero. [5 marks]