Air resistance kv² - horizontal motion or engine power

7 A particle \(P\) of mass \(m \mathrm {~kg}\) moves in a horizontal straight line against a resistive force of magnitude \(\mathrm { mkv } ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of \(P\) after it has moved a distance \(x \mathrm {~m}\) and \(k\) is a positive constant. The initial speed of \(P\) is \(u \mathrm {~ms} ^ { - 1 }\).

1. Show that \(\mathrm { x } = \frac { 1 } { \mathrm { k } } \ln 2\) when \(\mathrm { v } = \frac { 1 } { 2 } \mathrm { u }\).
   Beginning at the instant when the speed of \(P\) is \(\frac { 1 } { 2 } u\), an additional force acts on \(P\). This force has magnitude \(\frac { 5 \mathrm {~m} } { \mathrm { v } } \mathrm { N }\) and acts in the direction of increasing \(x\).
2. Show that when the speed of \(P\) has increased again to \(u \mathrm {~ms} ^ { - 1 }\), the total distance travelled by \(P\) is given by an expression of the form $$\frac { 1 } { 3 k } \ln \left( \frac { A - k u ^ { 3 } } { B - k u ^ { 3 } } \right) ,$$ stating the values of the constants \(A\) and \(B\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 A particle \(P\) of mass 0.4 kg is projected horizontally with velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on a smooth horizontal surface. The motion of \(P\) is opposed by a resisting force of magnitude \(0.2 v ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after projection.

1. Show that \(v = \frac { 8 } { 1 + 4 t }\).
2. Calculate the distance \(O P\) when \(t = 1.5\).

4 A particle of mass 0.5 kg moves along a horizontal straight line. Its velocity is \(v \mathrm {~ms} ^ { - 1 }\) at time \(t \mathrm {~s}\). The forces acting on the particle are a driving force of magnitude 50 N and a resistance of magnitude \(2 v ^ { 2 } \mathrm {~N}\). The initial velocity of the particle is \(3 \mathrm {~ms} ^ { - 1 }\).

1. Find an expression for \(v\) in terms of \(t\).
2. Deduce the limiting value of \(v\).

8 A car of mass 600 kg is driven along a straight horizontal road. The resistance to motion of the car is \(k v ^ { 2 }\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the car at time \(t\) seconds and \(k\) is a constant.

1. When the engine of the car has power 8 kW , show that the equation of motion of the car is $$600 \frac { \mathrm {~d} v } { \mathrm {~d} t } - \frac { 8000 } { v } + k v ^ { 2 } = 0$$
2. When the velocity of the car is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the engine is turned off.
   1. Show that the equation of motion of the car now becomes $$600 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - k v ^ { 2 }$$
   2. Find, in terms of \(k\), the time taken for the velocity of the car to drop to \(10 \mathrm {~ms} ^ { - 1 }\).

3 An aircraft of mass 80000 kg travelling at \(90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) touches down on a straight horizontal runway. It is brought to rest by braking and resistive forces which together are modelled by a horizontal force of magnitude ( \(27000 + 50 v ^ { 2 }\) ) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the aircraft. Find the distance travelled by the aircraft between touching down and coming to rest.

3. A cyclist and her bicycle have a combined mass of 75 kg . The cyclist travels along a straight horizontal road. The cyclist produces a constant driving force of magnitude 150 N . At time \(t\) seconds, the speed of the cyclist is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v < \sqrt { 50 }\). As the cyclist moves, the total resistance to motion of the cyclist and her bicycle has magnitude \(3 v ^ { 2 }\) newtons. The cyclist starts from rest. At time \(t\) seconds, she has travelled a distance \(x\) metres from her starting point. Find

1. \(v\) in terms of \(x\),
2. \(t\) in terms of \(v\).

1. A car of mass 500 kg moves along a straight horizontal road.

The engine of the car produces a constant driving force of 1800 N .
The car accelerates from rest from the fixed point \(O\) at time \(t = 0\) and at time \(t\) seconds the car is \(x\) metres from \(O\), moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car has magnitude \(2 v ^ { 2 } \mathrm {~N}\). At time \(T\) seconds, the car is at the point \(A\), moving with speed \(10 \mathrm {~ms} ^ { - 1 }\).

1. Show that \(T = \frac { 25 } { 6 } \ln 2\)
2. Show that the distance from \(O\) to \(A\) is \(125 \ln \frac { 9 } { 8 } \mathrm {~m}\).

5 When a car of mass 990 kg moves at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a horizontal straight road, the power of its engine is 8.8 kW .

1. Find the magnitude of the resistance to the motion of the car at this speed.
2. Assuming that the resistance has magnitude \(k v ^ { 2 }\) newtons when the speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of the constant \(k\). The power of the engine is now increased to 22 kW and remains constant at this value.
3. Using the model in part (ii), show that $$\frac { \mathrm { d } v } { \mathrm {~d} x } = \frac { 20000 - v ^ { 3 } } { 900 v ^ { 2 } } .$$
4. Hence show that the car moves about 300 m as its speed increases from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

A particle \(P\) of mass \(0.25\) kg is projected horizontally with speed \(5\) m s\(^{-1}\) from a fixed point \(O\) on a smooth horizontal surface and moves in a straight line on the surface. The only horizontal force acting on \(P\) has magnitude \(0.2v^2\) N, where \(v\) m s\(^{-1}\) is the velocity of \(P\) at time \(t\) s after it is projected from \(O\). This force is directed towards \(O\).

1. Find an expression for \(v\) in terms of \(t\). [5]

The particle \(P\) passes through a point \(X\) with speed \(0.2\) m s\(^{-1}\).

1. Find the average speed of \(P\) for its motion between \(O\) and \(X\). [5]

A lorry of mass \(M\) is moving along a straight horizontal road. The engine produces a constant driving force of magnitude \(F\). The total resistance to motion is modelled as having magnitude \(kv^2\), where \(k\) is a constant, and \(v\) is the speed of the lorry. Given the lorry moves with constant speed \(V\),

1. show that \(V = \sqrt{\frac{F}{k}}\). [2]

Given instead that the lorry starts from rest,

1. show that the distance travelled by the lorry in attaining a speed of \(\frac{1}{2}V\) is $$\frac{M}{2k}\ln\left(\frac{4}{3}\right).$$ [9]

A particle \(P\) of mass \(m\) moves along the positive \(x\)-axis. When its displacement from the origin \(O\) is \(x\) its velocity is \(v\), where \(v \geqslant 0\). It is subject to two forces: a constant force \(T\) in the positive \(x\) direction, and a resistive force which is proportional to \(v^2\).

1. Show that \(v^2 = \frac{1}{k}\left(T - Ae^{-\frac{2kx}{m}}\right)\) where \(A\) and \(k\) are constants. [5]

\(P\) starts from rest at \(O\).

1. Find an expression for the work done against the resistance to motion as \(P\) moves from \(O\) to the point where \(x = 1\). [4]
2. Find an expression for the limiting value of the velocity of \(P\) as \(x\) increases. [1]