6.06a Variable force: dv/dt or v*dv/dx methods

4 A small object of mass 0.4 kg is released from rest at a point 8 m above the ground. The object descends vertically and when its downwards displacement from its initial position is \(x \mathrm {~m}\) the object has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). While the object is moving, a force of magnitude \(0.2 v ^ { 2 } \mathrm {~N}\) opposes the motion.

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 10 - 0.5 v ^ { 2 }\).
2. Express \(v\) in terms of \(x\).
3. Find the increase in the value of \(v\) during the final 4 m of the descent of the object.

7 A particle \(P\) of mass 0.5 kg is attached to a fixed point \(O\) by a light elastic string of natural length 1 m and modulus of elasticity 16 N . The particle \(P\) is projected vertically upwards from \(O\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude \(0.1 x ^ { 2 } \mathrm {~N}\) acts on \(P\) when \(P\) has displacement \(x \mathrm {~m}\) above \(O\). After projection the upwards velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that, before the string becomes taut, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 10 - 0.2 x ^ { 2 }\).
2. Find the velocity of \(P\) at the instant the string becomes taut.
3. Find an expression for the acceleration of \(P\) while it is moving upwards after the string becomes taut.
4. Verify that \(P\) comes to instantaneous rest before the extension of the string is 0.5 m .
   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 0.5 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 16 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at the point 0.8 m vertically below \(O\). When the extension of the string is \(x \mathrm {~m}\), the downwards velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and a force of magnitude \(25 x ^ { 2 } \mathrm {~N}\) opposes the motion of \(P\).

1. Show that, when \(P\) is moving downwards, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 10 - 40 x - 50 x ^ { 2 }\).
2. For the instant when \(P\) has its greatest downwards speed, find the kinetic energy of \(P\) and the elastic potential energy stored in the string.

6 A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) on a plane inclined at \(30 ^ { \circ }\) to the horizontal. At time \(t \mathrm {~s}\) after its release, \(P\) has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and displacement \(x \mathrm {~m}\) down the plane from \(O\). The coefficient of friction between \(P\) and the plane increases as \(P\) moves down the plane, and equals \(0.1 x ^ { 2 }\).

1. Show that \(2 v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 10 - ( \sqrt { } 3 ) x ^ { 2 }\).
2. Calculate the maximum speed of \(P\).
3. Find the value of \(x\) at the point where \(P\) comes to rest.

7 A particle \(P\) is projected horizontally from a point \(O\) on a rough horizontal surface. The coefficient of friction between the particle and the surface is 0.2 . A horizontal force of magnitude \(0.06 t \mathrm {~N}\) directed away from \(O\) acts on \(P\), where \(t \mathrm {~s}\) is the time after projection. \(P\) comes to rest when \(t = 4\).

1. The particle begins to move again when \(t = 8\). Show that the mass of \(P\) is 0.24 kg .
2. Show that, for \(0 \leqslant t \leqslant 4 , \frac { \mathrm {~d} v } { \mathrm {~d} t } = 0.25 t - 2\), and find the speed of projection of \(P\).
3. Find the distance from \(O\) at which \(P\) comes to rest.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 \includegraphics[max width=\textwidth, alt={}, center]{fcf239a6-6558-43ec-b404-70aa349af6a9-3_177_880_1658_635} A particle \(P\) of mass \(\frac { 1 } { 10 } \mathrm {~kg}\) travels in a straight line on a smooth horizontal surface. It passes through the fixed point \(O\) with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t = 0\). After \(t\) seconds its displacement from \(O\) is \(x \mathrm {~m}\) and its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 } . P\) is subject to a single force of magnitude \(\frac { v } { 200 } \mathrm {~N}\) which acts in a direction opposite to the motion (see diagram).

1. Find an expression for \(v\) in terms of \(x\).
2. Find an expression for \(x\) in terms of \(t\).
3. Show that the value of \(x\) is less than 100 for all values of \(t\).
4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fcf239a6-6558-43ec-b404-70aa349af6a9-4_477_684_264_774} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Fig. 1 shows the cross section through the centre of mass \(C\) of a uniform L-shaped prism. \(C\) is \(x \mathrm {~cm}\) from \(O Y\) and \(y \mathrm {~cm}\) from \(O X\). Find the values of \(x\) and \(y\).
5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fcf239a6-6558-43ec-b404-70aa349af6a9-4_257_428_1064_902} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The prism is placed on a rough plane with \(O X\) in contact with the plane. The plane is tilted from the horizontal so that \(O X\) lies along a line of greatest slope, as shown in Fig. 2. When the angle of inclination of the plane is sufficiently great the prism starts to slide (without toppling). Show that the coefficient of friction between the prism and the plane is less than \(\frac { 7 } { 5 }\).
6. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fcf239a6-6558-43ec-b404-70aa349af6a9-4_303_414_1710_909} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} The prism is now placed on a rough plane with \(O Y\) in contact with the plane. The plane is tilted from the horizontal so that \(O Y\) lies along a line of greatest slope, as shown in Fig. 3. When the angle of inclination of the plane is sufficiently great the prism starts to topple (without sliding). Find the least possible value of the coefficient of friction between the prism and the plane. [3]

6 A cyclist and his machine have a total mass of 80 kg . The cyclist starts from rest and rides from the bottom to the top of a straight slope inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = 0.1\). The cyclist exerts a constant force of magnitude 120 N . There is a resisting force of magnitude \(8 v \mathrm {~N}\) acting on the cyclist, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the cyclist's speed at time \(t \mathrm {~s}\) after the start.

1. Show that \(\left( \frac { 1 } { 5 - v } \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 1 } { 10 }\).
2. Solve this differential equation and hence show that \(v = 5 \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 10 } t } \right)\).
3. Given that the cyclist takes 20 s to reach the top of the slope, find the length of the slope.

3 A car of mass 1000 kg is moving on a straight horizontal road. The driving force of the car is \(\frac { 28000 } { v } \mathrm {~N}\) and the resistance to motion is \(4 \nu \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car \(t\) seconds after it passes a fixed point on the road.

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 7000 - v ^ { 2 } } { 250 v }\). The car passes points \(A\) and \(B\) with speeds \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.
2. Find the time taken for the car to travel from \(A\) to \(B\).

7 A particle of mass 0.25 kg moves in a straight line on a smooth horizontal surface. A variable resisting force acts on the particle. At time \(t \mathrm {~s}\) the displacement of the particle from a point on the line is \(x \mathrm {~m}\), and its velocity is \(( 8 - 2 x ) \mathrm { m } \mathrm { s } ^ { - 1 }\). It is given that \(x = 0\) when \(t = 0\).

1. Find the acceleration of the particle in terms of \(x\), and hence find the magnitude of the resisting force when \(x = 1\).
2. Find an expression for \(x\) in terms of \(t\).
3. Show that the particle is always less than 4 m from its initial position.

7 A cyclist starts from rest at a point \(O\) and travels along a straight path. At time \(t \mathrm {~s}\) after starting, the displacement of the cyclist from \(O\) is \(x \mathrm {~m}\), and the acceleration of the cyclist is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 0.6 x ^ { 0.2 }\).

1. Find an expression for the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the cyclist in terms of \(x\).
2. Show that \(t = 2.5 x ^ { 0.4 }\).
3. Find the distance travelled by the cyclist in the first 10 s of the journey.

4 A particle of mass 0.4 kg is released from rest and falls vertically. A resisting force of magnitude \(0.08 v \mathrm {~N}\) acts upwards on the particle during its descent, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the particle at time \(t \mathrm {~s}\) after its release.

1. Show that the acceleration of the particle is \(( 10 - 0.2 v ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Find the velocity of the particle when \(t = 15\).

3 A particle \(P\) of mass 0.5 kg moves along the \(x\)-axis on a horizontal surface. When the displacement of \(P\) from the origin \(O\) is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Two horizontal forces act on \(P\); one force has magnitude \(\left( 1 + 0.3 x ^ { 2 } \right) \mathrm { N }\) and acts in the positive \(x\)-direction, and the other force has magnitude \(8 \mathrm { e } ^ { - x } \mathrm {~N}\) and acts in the negative \(x\)-direction.

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 2 + 0.6 x ^ { 2 } - 16 \mathrm { e } ^ { - x }\).
2. The velocity of \(P\) as it passes through \(O\) is \(6 \mathrm {~ms} ^ { - 1 }\). Find the velocity of \(P\) when \(x = 3\).
3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-3_259_745_278_740} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} A small sphere \(A\) of mass 0.15 kg is moving inside a fixed smooth hollow cylinder whose axis is vertical. \(A\) moves with constant speed \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle of radius 0.35 m , and is continuously in contact with both the plane base and the curved surface of the cylinder. Fig. 1 shows a vertical cross-section of the cylinder through its axis. Find the magnitude of the force exerted on \(A\) by

7 A particle \(P\) of mass 0.1 kg is projected vertically upwards from a point \(O\) with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Air resistance of magnitude \(0.1 v \mathrm {~N}\) opposes the motion, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\) at time \(t \mathrm {~s}\) after projection.

1. Show that, while \(P\) is moving upwards, \(\frac { 1 } { v + 10 } \frac { \mathrm {~d} v } { \mathrm {~d} t } = - 1\).
2. Hence find an expression for \(v\) in terms of \(t\), and explain why it is valid only for \(0 \leqslant t \leqslant \ln 3\).
3. Find the initial acceleration of \(P\).

7 A particle \(P\) of mass 0.3 kg is projected vertically upwards from the ground with an initial speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(P\) is at height \(x \mathrm {~m}\) above the ground, its upward speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It is given that $$3 v - 90 \ln ( v + 30 ) + x = A ,$$ where \(A\) is a constant.

1. Differentiate this equation with respect to \(x\) and hence show that the acceleration of the particle is \(- \frac { 1 } { 3 } ( v + 30 ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Find, in terms of \(v\), the resisting force acting on the particle.
3. Find the time taken for \(P\) to reach its maximum height.

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.

3 A particle \(P\) is moving in a horizontal straight line. Initially \(P\) is at the point \(O\) on the line and is moving with velocity \(25 \mathrm {~ms} ^ { - 1 }\). At time \(t \mathrm {~s}\) after passing through \(O\), the acceleration of \(P\) is \(\frac { 4000 } { ( 5 t + 4 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) in the direction \(P O\). The displacement of \(P\) from \(O\) at time \(t\) is \(x \mathrm {~m}\). Find an expression for \(x\) in terms of \(t\). \includegraphics[max width=\textwidth, alt={}, center]{c486c59a-2493-4dd3-bf1e-dde57fe744d9-06_894_809_260_628} An object is composed of a hemispherical shell of radius \(2 a\) attached to a closed hollow circular cylinder of height \(h\) and base radius \(a\). The hemispherical shell and the hollow cylinder are made of the same uniform material. The axes of symmetry of the shell and the cylinder coincide. \(A B\) is a diameter of the lower end of the cylinder (see diagram).

1. Find, in terms of \(a\) and \(h\), an expression for the distance of the centre of mass of the object from \(A B\). [4]
   The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 2 } { 3 }\). The object is in equilibrium with \(A B\) in contact with the plane and lying along a line of greatest slope of the plane.
2. Find the set of possible values of \(h\), in terms of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{c486c59a-2493-4dd3-bf1e-dde57fe744d9-08_629_1358_269_367} A light inextensible string \(A B\) passes through two small holes \(C\) and \(D\) in a smooth horizontal table where \(A C = 3 a\) and \(D B = a\). A particle of mass \(m\) is attached at the end \(A\) and moves in a horizontal circle with angular velocity \(\omega\). A particle of mass \(\frac { 3 } { 4 } m\) is attached to the end \(B\) and moves in a horizontal circle with angular velocity \(k \omega\). \(A C\) makes an angle \(\theta\) with the downward vertical and \(D B\) makes an angle \(\theta\) with the horizontal (see diagram). Find the value of \(k\).

5 A particle \(P\) of mass 4 kg is moving in a horizontal straight line. At time \(t\) s the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) and the displacement of \(P\) from a fixed point \(O\) on the line is \(x \mathrm {~m}\). The only force acting on \(P\) is a resistive force of magnitude \(\left( 4 \mathrm { e } ^ { - x } + 12 \right) \mathrm { e } ^ { - x } \mathrm {~N}\). When \(\mathrm { t } = 0 , \mathrm { x } = 0\) and \(v = 4\).

1. Show by integration that \(\mathrm { v } = \frac { 1 + 3 \mathrm { e } ^ { \mathrm { x } } } { \mathrm { e } ^ { \mathrm { x } } }\).
2. Find an expression for \(x\) in terms of \(t\). \includegraphics[max width=\textwidth, alt={}, center]{ad8b126c-d739-4e2a-8ce3-7811a61f5876-10_510_889_269_580} \(A B\) and \(B C\) are two fixed smooth vertical barriers on a smooth horizontal surface, with angle \(\mathrm { ABC } = 60 ^ { \circ }\). A particle of mass \(m\) is moving with speed \(u\) on the surface. The particle strikes \(A B\) at an angle \(\theta\) with \(A B\). It then strikes \(B C\) and rebounds at an angle \(\beta\) with \(B C\) (see diagram). The coefficient of restitution between the particle and each barrier is \(e\) and \(\tan \theta = 2\). The kinetic energy of the particle after the first collision is \(40 \%\) of its kinetic energy before the first collision.
   1. Find the value of \(e\).
   2. Find the size of angle \(\beta\). \includegraphics[max width=\textwidth, alt={}, center]{ad8b126c-d739-4e2a-8ce3-7811a61f5876-12_965_1059_267_502} A uniform cylinder with a rough surface and of radius \(a\) is fixed with its axis horizontal. Two identical uniform rods \(A B\) and \(B C\), each of weight \(W\) and length \(2 a\), are rigidly joined at \(B\) with \(A B\) perpendicular to \(B C\). The rods rest on the cylinder in a vertical plane perpendicular to the axis of the cylinder with \(A B\) at an angle \(\theta\) to the horizontal. \(D\) and \(E\) are the midpoints of \(A B\) and \(B C\) respectively and also the points of contact of the rods with the cylinder (see diagram). The rods are about to slip in a clockwise direction. The coefficient of friction between each rod and the cylinder is \(\mu\). The normal reaction between \(A B\) and the cylinder is \(R\) and the normal reaction between \(B C\) and the cylinder is \(N\).
   3. Find the ratio \(R : N\) in terms of \(\mu\).
   4. Given that \(\mu = \frac { 1 } { 3 }\), find the value of \(\tan \theta\).
      If you use the following page to complete the answer to any question, the question number must be clearly shown.

6 A cyclist and his bicycle have a total mass of 81 kg . The cyclist starts from rest and rides in a straight line. The cyclist exerts a constant force of 135 N and the motion is opposed by a resistance of magnitude \(9 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the cyclist's speed at time \(t \mathrm {~s}\) after starting.

1. Show that \(\frac { 9 } { 15 - v } \frac { \mathrm {~d} v } { \mathrm {~d} t } = 1\).
2. Solve this differential equation to show that \(v = 15 \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 9 } t } \right)\).
3. Find the distance travelled by the cyclist in the first 9 s of the motion.

5 A ball of mass 0.05 kg is released from rest at a height \(h \mathrm {~m}\) above the ground. At time \(t \mathrm {~s}\) after its release, the downward velocity of the ball is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Air resistance opposes the motion of the ball with a force of magnitude \(0.01 \nu \mathrm {~N}\).

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = 10 - 0.2 v\). Hence find \(v\) in terms of \(t\).
2. Given that the ball reaches the ground when \(t = 2\), calculate \(h\).

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 \(t = 1.5\).

3 A particle \(P\) of mass 0.2 kg is released from rest and falls vertically. At time \(t \mathrm {~s}\) after release \(P\) has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude \(0.8 v \mathrm {~N}\) acts on \(P\).

1. Show that the acceleration of \(P\) is \(( 10 - 4 v ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Find the value of \(v\) when \(t = 0.6\).

6 \includegraphics[max width=\textwidth, alt={}, center]{e30ba526-db21-4904-96dc-c12a1f67c81a-4_238_725_258_712} Two particles \(P\) and \(Q\), of masses 0.4 kg and 0.2 kg respectively, are attached to opposite ends of a light inextensible string. \(P\) is placed on a horizontal table and the string passes over a small smooth pulley at the edge of the table. The string is taut and the part of the string attached to \(Q\) is vertical (see diagram). The coefficient of friction between \(P\) and the table is 0.5 . \(Q\) is projected vertically downwards with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and at time \(t \mathrm {~s}\) after the instant of projection the speed of the particles is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The motion of each particle is opposed by a resisting force of magnitude \(0.9 v \mathrm {~N}\). The particle \(P\) does not reach the pulley.

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 3 v\).
2. Find the value of \(t\) when the particles have speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the distance that each particle has travelled in this time.

3 A particle \(P\) of mass 0.8 kg moves along the \(x\)-axis on a horizontal surface. When the displacement of \(P\) from the origin \(O\) is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Two horizontal forces act on \(P\). One force has magnitude \(4 \mathrm { e } ^ { - x } \mathrm {~N}\) and acts in the positive \(x\)-direction. The other force has magnitude \(2.4 x ^ { 2 } \mathrm {~N}\) and acts in the negative \(x\)-direction.

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 \mathrm { e } ^ { - x } - 3 x ^ { 2 }\).
2. The velocity of \(P\) as it passes through \(O\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the velocity of \(P\) when \(x = 2\).

3 A small ball of mass \(m \mathrm {~kg}\) is projected vertically upwards with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) upwards when it is \(x \mathrm {~m}\) above the point of projection. A resisting force of magnitude \(0.02 m v \mathrm {~N}\) acts on the ball during its upward motion.

1. Show that, while the ball is moving upwards, \(\left( \frac { 500 } { v + 500 } - 1 \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 0.02\).
2. Find the greatest height of the ball above its point of projection.

5 A particle \(P\) of mass 0.5 kg is projected vertically upwards from a point on a horizontal surface. A resisting force of magnitude \(0.02 v ^ { 2 } \mathrm {~N}\) acts on \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the upward velocity of \(P\) when it is a height of \(x \mathrm {~m}\) above the surface. The initial speed of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that, while \(P\) is moving upwards, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 10 - 0.04 v ^ { 2 }\).
2. Find the greatest height of \(P\) above the surface.
3. Find the speed of \(P\) immediately before it strikes the surface after descending.