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

7 A stone of mass \(m\) is moving along the smooth horizontal floor of a tank which is filled with a viscous liquid. At time \(t\), the stone has speed \(v\). As the stone moves, it experiences a resistance force of magnitude \(\lambda m v\), where \(\lambda\) is a constant.

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - \lambda v$$
2. The initial speed of the stone is \(U\). Show that $$v = U \mathrm { e } ^ { - \lambda t }$$

5 A particle is moving along a straight line. At time \(t\), the velocity of the particle is \(v\). The acceleration of the particle throughout the motion is \(- \frac { \lambda } { v ^ { \frac { 1 } { 4 } } }\), where \(\lambda\) is a positive constant. The velocity of the particle is \(u\) when \(t = 0\). Find \(v\) in terms of \(u , \lambda\) and \(t\).
(7 marks)

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)

7 A stone, of mass 5 kg , is projected vertically downwards, in a viscous liquid, with an initial speed of \(7 \mathrm {~ms} ^ { - 1 }\). At time \(t\) seconds after it is projected, the stone has speed \(v \mathrm {~ms} ^ { - 1 }\) and it experiences a resistance force of magnitude \(9.8 v\) newtons.

1. When \(t \geqslant 0\), show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 1.96 ( v - 5 )$$ (2 marks)
2. Find \(v\) in terms of \(t\).

6 A car accelerates from rest along a straight horizontal road. The car's engine produces a constant horizontal force of magnitude 4000 N .
At time \(t\) seconds, the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and a resistance force of magnitude \(40 v\) newtons acts upon the car. The mass of the car is 1600 kg .

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 100 - v } { 40 }\).
2. Find the velocity of the car at time \(t\).

7 A parachutist, of mass 72 kg , is falling vertically. He opens his parachute at time \(t = 0\) when his speed is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He then experiences an air resistance force of magnitude \(240 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is his speed at time \(t\) seconds.

1. When \(t > 0\), show that \(- \frac { 3 } { 10 } \frac { \mathrm {~d} v } { \mathrm {~d} t } = v - 2.94\).
2. Find \(v\) in terms of \(t\).
3. Sketch a graph to show how, for \(t \geqslant 0\), the parachutist's speed varies with time.
   [0pt] [2 marks]

8 Carol, a bungee jumper of mass 70 kg , is attached to one end of a light elastic cord of natural length 26 metres and modulus of elasticity 1456 N . The other end of the cord is attached to a fixed horizontal platform which is at a height of 69 metres above the ground. Carol steps off the platform at the point where the cord is attached and falls vertically. Hooke's law can be assumed to apply whilst the cord is taut. Model Carol as a particle and assume air resistance to be negligible.
When Carol has fallen \(x \mathrm {~m}\), her speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By considering energy, show that $$5 v ^ { 2 } = 306 x - 4 x ^ { 2 } - 2704 \text { for } x \geqslant 26$$
2. Why is the expression found in part (a) not true when \(x\) takes values less than 26?
3. Find the maximum value of \(x\).
4. 1. Find the distance fallen by Carol when her speed is a maximum.
   2. Hence find Carol's maximum speed.

1 \includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-2_246_693_278_731} A particle \(P\) of mass 0.4 kg moving in a straight line has speed \(8.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). An impulse applied to \(P\) deflects it through \(45 ^ { \circ }\) and reduces its speed to \(5.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Calculate the magnitude and direction of the impulse exerted on \(P\). \(2 \quad O\) is a fixed point on a horizontal straight line. A particle \(P\) of mass 0.5 kg is released from rest at \(O\). At time \(t\) seconds after release the only force acting on \(P\) has magnitude \(\left( 1 + k t ^ { 2 } \right) \mathrm { N }\) and acts horizontally and away from \(O\) along the line, where \(k\) is a positive constant.

1. Find the speed of \(P\) in terms of \(k\) and \(t\).
2. Given that \(P\) is 2 m from \(O\) when \(t = 1\), find the value of \(k\) and the time taken by \(P\) to travel 20 m from \(O\).

5 The pilot of a hot air balloon keeps it at a fixed altitude by dropping sand from the balloon. Each grain of sand has mass \(m \mathrm {~kg}\) and is released from rest. When a grain has fallen a distance \(x \mathrm {~m}\), it has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Each grain falls vertically and the only forces acting on it are its weight and air resistance of magnitude \(m k v ^ { 2 } \mathrm {~N}\), where \(k\) is a positive constant.

1. Show that \(\left( \frac { v } { g - k v ^ { 2 } } \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 1\).
2. Find \(v ^ { 2 }\) in terms of \(k , g\) and \(x\). Hence show that, as \(x\) becomes large, the limiting value of \(v\) is \(\sqrt { \frac { g } { k } }\).
3. Given that the altitude of the balloon is 300 m and that each grain strikes the ground at \(90 \%\) of its limiting velocity, find \(k\).

4 A particle of mass \(m \mathrm {~kg}\) is released from rest at a fixed point \(O\) and falls vertically. The particle is subject to an upward resisting force of magnitude \(0.49 m v \mathrm {~N}\) where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the particle when it has fallen a distance of \(x \mathrm {~m}\) from \(O\).

1. Write down a differential equation for the motion of the particle, and show that the equation can be written as \(\left( \frac { 20 } { 20 - v } - 1 \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 0.49\).
2. Hence find an expression for \(x\) in terms of \(v\).

6 A stone of mass 0.125 kg falls freely under gravity, from rest, until it has travelled a distance of 10 m . The stone then continues to fall in a medium which exerts an upward resisting force of \(0.025 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the stone \(t \mathrm {~s}\) after the instant that it enters the resisting medium.

1. Show by integration that \(v = 49 - 35 \mathrm { e } ^ { - 0.2 t }\).
2. Find how far the stone travels during the first 3 seconds in the medium.

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 \(x = 1\).

7 A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) and falls vertically. Air resistance of magnitude \(\frac { v ^ { 2 } } { 2000 } \mathrm {~N}\) acts upwards on \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) when it has fallen a distance of \(x \mathrm {~m}\).

1. Show that \(\left( \frac { 400 v } { 3920 - v ^ { 2 } } \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 1\).
2. Find \(v ^ { 2 }\) in terms of \(x\) and hence show that \(v ^ { 2 } < 3920\) for all values of \(x\).
3. Find the work done against the air resistance while \(P\) is falling, from \(O\), to the point where its downward acceleration is \(5.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). }{www.ocr.org.uk}) after the live examination series.
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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 \(u\).

3 At time \(t = 0 \mathrm {~s}\) a particle \(P\), of mass 0.3 kg , is 1 m away from a point \(O\) on a smooth horizontal plane and is moving away from \(O\) with speed \(\sqrt { 5 } \mathrm {~ms} ^ { - 1 }\). The only horizontal force acting on \(P\) has magnitude \(1.5 x \mathrm {~N}\), where \(x\) is the distance \(O P\), and acts away from \(O\).

1. Show that the speed of \(P , v \mathrm {~ms} ^ { - 1 }\), is given by \(v = \sqrt { 5 } x\).
2. Find an expression for \(v\) in terms of \(t\).

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.

3 A particle \(P\) of mass 0.2 kg is projected horizontally with speed \(u \mathrm {~ms} ^ { - 1 }\) from a fixed point \(O\) on a smooth horizontal surface. \(P\) moves in a straight line and, at time \(t \mathrm {~s}\) after projection, \(P\) has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is \(x \mathrm {~m}\) from \(O\). The only force acting on \(P\) has magnitude \(0.4 v ^ { 2 } \mathrm {~N}\) and is directed towards \(O\).

1. Show that \(\frac { 1 } { v } \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 2\).
2. Hence show that \(v = u \mathrm { e } ^ { - 2 x }\).
3. Find \(u\), given that \(x = 2\) when \(t = 4\).

4 A particle \(P\) of mass \(m \mathrm {~kg}\) is held at rest at a point \(O\) on a fixed plane inclined at an angle \(\sin ^ { - 1 } \left( \frac { 4 } { 7 } \right)\) to the horizontal. \(P\) is released and moves down the plane. The total resistance acting on \(P\) is \(0.2 m v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\).

1. Show that \(5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = 28 - v\) and hence find an expression for \(v\) in terms of \(t\).
2. Find the acceleration of \(P\) when \(t = 10\).

3 A particle \(P\) of mass \(m \mathrm {~kg}\) is released from rest and falls vertically. When \(P\) has fallen a distance of \(x \mathrm {~m}\) it has a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The only forces acting on \(P\) are its weight and air resistance of magnitude \(\frac { 1 } { 400 } m v ^ { 2 } \mathrm {~N}\).

1. Find \(v ^ { 2 }\) in terms of \(x\) and show that \(v ^ { 2 }\) must be less than 3920 .
2. Find the speed of \(P\) when it has fallen 100 m .

4 A particle \(P\) of mass 0.4 kg is projected horizontally with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 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 away from \(O\) with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a force of magnitude \(1.6 v ^ { 2 } \mathrm {~N}\) 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 \(v = 2 \mathrm { e } ^ { - 4 x }\).
2. Find the distance travelled by \(P\) in the 0.5 seconds after it leaves \(O\).

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.

1 The fixed point A is at a height \(4 b\) above a smooth horizontal surface, and C is the point on the surface which is vertically below A. A light elastic string, of natural length \(3 b\) and modulus of elasticity \(\lambda\), has one end attached to A and the other end attached to a block of mass \(m\). The block is released from rest at a point B on the surface where \(\mathrm { BC } = 3 b\), as shown in Fig. 1. You are given that the block remains on the surface and moves along the line BC . \begin{figure}[h] \end{figure}

1. Show that immediately after release the acceleration of the block is \(\frac { 2 \lambda } { 5 m }\).
2. Show that, when the block reaches C , its speed \(v\) is given by \(v ^ { 2 } = \frac { \lambda b } { m }\).
3. Show that the equation \(v ^ { 2 } = \frac { \lambda b } { m }\) is dimensionally consistent. The time taken for the block to move from B to C is given by \(k m ^ { \alpha } b ^ { \beta } \lambda ^ { \gamma }\), where \(k\) is a dimensionless constant.
4. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\). When the string has natural length 1.2 m and modulus of elasticity 125 N , and the block has mass 8 kg , the time taken for the block to move from B to C is 0.718 s .
5. Find the time taken for the block to move from B to C when the string has natural length 9 m and modulus of elasticity 20 N , and the block has mass 75 kg .

2. A particle \(P\) of mass 0.5 kg moves along the positive \(x\)-axis under the action of a single force directed away from the origin \(O\). When \(P\) is \(x\) metres from \(O\), the magnitude of the force is \(3 x ^ { \frac { 1 } { 2 } } \mathrm {~N}\) and \(P\) has a speed of \(v \mathrm {~ms} ^ { - 1 }\). Given that when \(x = 1 , P\) is moving away from \(O\) with speed \(2 \mathrm {~ms} ^ { - 1 }\),

1. find an expression for \(v ^ { 2 }\) in terms of \(x\),
2. show that when \(x = 4 , P\) has a speed of \(7.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 1 decimal place.

4. Whilst in free-fall a parachutist falls vertically such that his velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when he is \(x\) metres below his initial position is given by $$v ^ { 2 } = k g \left( 1 - \mathrm { e } ^ { - \frac { 2 x } { k } } \right) ,$$ where \(k\) is a constant.
Given that he experiences an acceleration of \(\mathrm { f } \mathrm { m } \mathrm { s } ^ { - 2 }\),

1. show that \(f = g \mathrm { e } ^ { - \frac { 2 x } { k } }\). After falling a large distance, his velocity is constant at \(49 \mathrm {~ms} ^ { - 1 }\).
2. Find the value of \(k\).
3. Hence, express \(f\) in the form ( \(\lambda - \mu v ^ { 2 }\) ) where \(\lambda\) and \(\mu\) are constants which you should find.
   (4 marks)

7. A particle is travelling along the \(x\)-axis. At time \(t = 0\), the particle is at \(O\) and it travels such that its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at a distance \(x\) metres from \(O\) is given by $$v = \frac { 2 } { x + 1 }$$ The acceleration of the particle is \(a \mathrm {~ms} ^ { - 2 }\).

1. Show that \(a = \frac { - 4 } { ( x + 1 ) ^ { 3 } }\).
   (4 marks) The points \(A\) and \(B\) lie on the \(x\)-axis. Given that the particle travels \(d\) metres from \(O\) to \(A\) in \(T\) seconds and 4 metres from \(A\) to \(B\) in 9 seconds,
2. show that \(d = 1.5\),
3. find \(T\).