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

6 \includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-4_182_844_264_653} A particle \(P\) of mass 0.4 kg travels on a horizontal surface along the line \(O A\) in the direction from \(O\) to \(A\). 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 it passes through the fixed point \(O\) (see diagram). The speed of \(P\) at \(O\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Assume that the horizontal surface is smooth. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} x } = - \frac { 1 } { 4 }\), where \(x \mathrm {~m}\) is the distance of \(P\) from \(O\) at time \(t \mathrm {~s}\), and hence find the distance from \(O\) at which the speed of \(P\) is zero.
2. Assume instead that the horizontal surface is not smooth and that the coefficient of friction between \(P\) and the surface is \(\frac { 3 } { 40 }\).
   1. Show that \(4 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - ( v + 3 )\).
   2. Hence find the value of \(t\) for which the speed of \(P\) is zero.

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 \(x = 0\).

3 \includegraphics[max width=\textwidth, alt={}, center]{835616aa-0b2b-4e8c-bbbf-60b72dc5ea3e-2_145_792_1656_680} A particle \(P\) of mass 0.6 kg moves in a straight line on a smooth horizontal surface. A force of magnitude \(\frac { 3 } { x ^ { 3 } }\) newtons acts on the particle in the direction from \(P\) to \(O\), where \(O\) is a fixed point of the surface and \(x \mathrm {~m}\) is the distance \(O P\) (see diagram). The particle \(P\) is released from rest at the point where \(x = 10\). Find the speed of \(P\) when \(x = 2.5\).

5 The acceleration of a particle moving in a straight line is \(( x - 2.4 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) when its displacement from a fixed point \(O\) of the line is \(x \mathrm {~m}\). The velocity of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and it is given that \(v = 2.5\) when \(x = 0\). Find

1. an expression for \(v\) in terms of \(x\),
2. the minimum value of \(v\).

4 An object of mass 0.4 kg is projected vertically upwards from the ground, with an initial speed of \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude \(0.1 v\) newtons acts on the object during its ascent, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the object at time \(t \mathrm {~s}\) after it starts to move.

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 0.25 ( v + 40 )\).
2. Find the value of \(t\) at the instant that the object reaches its maximum height.

2 A particle starts from rest at \(O\) and travels in a straight line. Its acceleration is \(( 3 - 2 x ) \mathrm { ms } ^ { - 2 }\), where \(x \mathrm {~m}\) is the displacement of the particle from \(O\).

1. Find the value of \(x\) for which the velocity of the particle reaches its maximum value.
2. Find this maximum velocity.

6 One end of a light elastic string of natural length 1.25 m and modulus of elasticity 20 N is attached to a fixed point \(O\). A particle \(P\) of mass 0.5 kg is attached to the other end of the string. \(P\) is held at rest at \(O\) and then released. When the extension of the string is \(x \mathrm {~m}\) the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = - 32 x ^ { 2 } + 20 x + 25\).
2. Find the maximum speed of \(P\).
3. Find the acceleration of \(P\) when it is at its lowest point.

7 A particle \(P\) of mass 0.5 kg moves on a horizontal surface along the straight line \(O A\), in the direction from \(O\) to \(A\). The coefficient of friction between \(P\) and the surface is 0.08 . Air resistance of magnitude \(0.2 v \mathrm {~N}\) opposes the motion, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\) at time \(t \mathrm {~s}\). The particle passes through \(O\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 0\).

1. Show that \(2.5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - ( v + 2 )\) and hence find the value of \(t\) when \(v = 0\).
2. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 6 \mathrm { e } ^ { - 0.4 t } - 2\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\) at time \(t \mathrm {~s}\), and hence find the distance \(O P\) when \(v = 0\).

3 A particle \(P\) starts from a fixed point \(O\) and moves in a straight line. When the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\), its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its acceleration is \(\frac { 1 } { x + 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Given that \(v = 2\) when \(x = 0\), use integration to show that \(v ^ { 2 } = 2 \ln \left( \frac { 1 } { 2 } x + 1 \right) + 4\).
2. Find the value of \(v\) when the acceleration of \(P\) is \(\frac { 1 } { 4 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

7 A particle \(P\) of mass 0.25 kg moves in a straight line on a smooth horizontal surface. \(P\) starts at the point \(O\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves towards a fixed point \(A\) on the line. At time \(t \mathrm {~s}\) the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) and the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resistive force of magnitude (5-x) N acts on \(P\) in the direction towards \(O\).

1. Form a differential equation in \(v\) and \(x\). By solving this differential equation, show that \(v = 10 - 2 x\).
2. Find \(x\) in terms of \(t\), and hence show that the particle is always less than 5 m from \(O\).

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 \(t = 0.5\).

4 A particle \(P\) starts from rest at a point \(O\) and travels in a straight line. The acceleration of \(P\) is \(( 15 - 6 x ) \mathrm { m } \mathrm { s } ^ { - 2 }\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\).

1. Find the value of \(x\) for which \(P\) reaches its maximum velocity, and calculate this maximum velocity.
2. Calculate the acceleration of \(P\) when it is at instantaneous rest and \(x > 0\).

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.

6 \includegraphics[max width=\textwidth, alt={}, center]{9d377c95-09b8-4893-b29f-8517a5016e8b-3_151_949_1206_598} \(O\) and \(A\) are fixed points on a horizontal surface, with \(O A = 0.5 \mathrm {~m}\). A particle \(P\) of mass 0.2 kg is projected horizontally with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) in the direction \(O A\) and moves in a straight line (see diagram). At time \(t \mathrm {~s}\) after projection, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its displacement from \(O\) is \(x \mathrm {~m}\). The coefficient of friction between the surface and \(P\) is 0.5 , and a force of magnitude \(\frac { 0.4 } { x ^ { 2 } } \mathrm {~N}\) acts on \(P\) in the direction \(P O\).

1. Show that, while the particle is in motion, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - \left( 5 + \frac { 2 } { x ^ { 2 } } \right)\).
2. Calculate the distance travelled by \(P\) before it comes to rest, and show that \(P\) does not subsequently move.

5 A particle \(P\) of mass 0.4 kg is released from rest at the top of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The motion of \(P\) down the slope is opposed by a force of magnitude \(0.6 x \mathrm {~N}\), where \(x \mathrm {~m}\) is the distance \(P\) has travelled down the slope. \(P\) comes to rest before reaching the foot of the slope. Calculate

1. the greatest speed of \(P\) during its motion,
2. the distance travelled by \(P\) during its motion.

1 A particle \(P\) of mass 0.6 kg is projected horizontally with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on a smooth horizontal surface. A horizontal force of magnitude \(0.3 x \mathrm {~N}\) acts on \(P\) in the direction \(O P\), where \(x \mathrm {~m}\) is the distance of \(P\) from \(O\). Calculate the velocity of \(P\) when \(x = 8\).

4 A particle \(P\) of mass 0.25 kg moves in a straight line on a smooth horizontal surface. At time \(t \mathrm {~s}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A variable force of magnitude \(3 t \mathrm {~N}\) opposes the motion of \(P\).

1. Given that \(P\) comes to rest when \(t = 3\), find \(v\) when \(t = 0\).
2. Calculate the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 3\).

3 A particle \(P\) of mass 0.2 kg is projected horizontally from a fixed point \(O\), and moves in a straight line on a smooth horizontal surface. A force of magnitude \(0.4 x \mathrm {~N}\) acts on \(P\) in the direction \(P O\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\).

1. Given that \(P\) comes to instantaneous rest when \(x = 2.5\), find the initial kinetic energy of \(P\).
2. Find the value of \(x\) on the first occasion when the speed of \(P\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

4 A particle of mass 0.2 kg is projected vertically downwards with initial speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude \(0.09 v \mathrm {~N}\) acts vertically upwards on the particle during its descent, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the downwards velocity of the particle at time \(t \mathrm {~s}\) after being set in motion.

1. Show that the acceleration of the particle is \(( 10 - 0.45 v ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Find \(v\) when \(t = 1.5\).

7 A particle \(P\) of mass 0.5 kg moves in a straight line on a smooth horizontal surface. The velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\). A single horizontal force of magnitude \(0.16 \mathrm { e } ^ { x } \mathrm {~N}\) acts on \(P\) in the direction \(O P\). The velocity of \(P\) when it is at \(O\) is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v = 0.8 \mathrm { e } ^ { \frac { 1 } { 2 } x }\).
2. Find the time taken by \(P\) to travel 1.4 m from \(O\).

6 A particle \(P\) of mass 0.6 kg is released from rest at a point above ground level and falls vertically. The motion of \(P\) is opposed by a force of magnitude \(3 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\). Immediately before \(P\) reaches the ground, \(v = 1.95\).

1. Calculate the time after its release when \(P\) reaches the ground. \(P\) is now projected horizontally with speed \(1.95 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) across a smooth horizontal surface. The motion of \(P\) is again opposed by a force of magnitude \(3 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\).
2. Calculate the distance \(P\) travels after projection before coming to rest.

3 A small block \(B\) of mass 0.2 kg is placed at a fixed point \(O\) on a smooth horizontal surface. A horizontal force of magnitude 0.42 N is applied to \(B\). At time \(t \mathrm {~s}\) after the force is first applied, the velocity of \(B\) away from \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the value of \(v\) when \(t = 1\). For \(t > 1\) an additional force, of magnitude \(0.32 t \mathrm {~N}\) and directed towards \(O\), is applied to \(B\). The force of magnitude 0.42 N continues to act as before.
2. Find the value of \(v\) when \(t = 2\). For \(t > 2\) a third force, of magnitude \(0.06 t ^ { 2 } \mathrm {~N}\) and directed away from \(O\), is applied to \(B\). The other two forces continue to act as before.
3. Show that the velocity of \(B\) is the same when \(t = 2\) and when \(t = 3\).

7 A force of magnitude \(0.4 t \mathrm {~N}\), applied at an angle of \(30 ^ { \circ }\) above the horizontal, acts on a particle \(P\), where \(t \mathrm {~s}\) is the time since the force starts to act. \(P\) is at rest on rough horizontal ground when \(t = 0\). The mass of \(P\) is 0.2 kg and the coefficient of friction between \(P\) and the ground is \(\mu\).

1. Given that \(P\) is about to slip when \(t = 2\), find \(\mu\) and the value of \(t\) for the instant when \(P\) loses contact with the ground.
2. While \(P\) is moving on the ground, it has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\). Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 2.165 t - 4.330$$ where the coefficients are correct to 4 significant figures.
3. Calculate the speed of \(P\) when it loses contact with the ground. {www.cie.org.uk} after the live examination series. }

5 \includegraphics[max width=\textwidth, alt={}, center]{8f8492a7-8a83-4eb2-81ee-99b4a385b704-3_876_483_260_840} A uniform triangular prism of weight 20 N rests on a horizontal table. \(A B C\) is the cross-section through the centre of mass of the prism, where \(B C = 0.5 \mathrm {~m} , A B = 0.4 \mathrm {~m} , A C = 0.3 \mathrm {~m}\) and angle \(B A C = 90 ^ { \circ }\). The vertical plane \(A B C\) is perpendicular to the edge of the table. The point \(D\) on \(A C\) is at the edge of the table, and \(A D = 0.25 \mathrm {~m}\). One end of a light elastic string of natural length 0.6 m and modulus of elasticity 48 N is attached to \(C\) and a particle of mass 2.5 kg is attached to the other end of the string. The particle is released from rest at \(C\) and falls vertically (see diagram).

1. Show that the tension in the string is 60 N at the instant when the prism topples.
2. Calculate the speed of the particle at the instant when the prism topples.

5 A particle \(P\) of mass 0.4 kg is placed at rest at a point \(A\) on a rough horizontal surface. A horizontal force, directed away from \(A\) and with magnitude \(0.6 t \mathrm {~N}\), acts on \(P\), where \(t \mathrm {~s}\) is the time after \(P\) is placed at \(A\). The coefficient of friction between \(P\) and the surface is 0.3 , and \(P\) has displacement from \(A\) of \(x \mathrm {~m}\) at time \(t \mathrm {~s}\).

1. Show that \(P\) starts to move when \(t = 2\). Show also that when \(P\) is in motion it has acceleration \(( 1.5 t - 3 ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Express the velocity of \(P\) in terms of \(t\), for \(t \geqslant 2\).
3. Express \(x\) in terms of \(t\), for \(t \geqslant 2\).