Air resistance kv - vertical motion

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.

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 \(t = 1.5\).

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 \(t = 15\).

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 \(P\).

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 \(h\).

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.

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 \(m \mathrm {~kg}\) 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. At time \(t = 0\), a particle of mass \(m\) is projected vertically downwards with speed \(U\) from a point above the ground. At time \(t\) the speed of the particle is \(v\) and the magnitude of the air resistance is modelled as being \(m k v\), where \(k\) is a constant. Given that \(U < \frac { \boldsymbol { g } } { \mathbf { 2 } \boldsymbol { k } }\), find, in terms of \(k , U\) and \(g\), the time taken for the particle to double its speed.
(8)

A particle \(P\) of mass \(m\) kg is projected vertically upwards from a point \(O\), with speed \(20\) m s\(^{-1}\), and moves under gravity. There is a resistive force of magnitude \(2mv\) N, where \(v\) m s\(^{-1}\) is the speed of \(P\) at time \(t\) s after projection.

1. Find an expression for \(v\) in terms of \(t\), while \(P\) is moving upwards. [6]

A particle of mass \(m\) kg falls vertically under gravity, from rest. At time \(t\) s, \(P\) has fallen \(x\) m and has velocity \(v\) m s\(^{-1}\). The only forces acting on \(P\) are its weight and a resistance of magnitude \(kmgv\) N, where \(k\) is a constant.

1. Find an expression for \(v\) in terms of \(t\), \(g\) and \(k\). [5]
2. Given that \(k = 0.05\), find, in metres, how far \(P\) has fallen when its speed is \(12\) m s\(^{-1}\). [5]

A stone, of mass \(m\), falls vertically downwards under gravity through still water. At time \(t\), the stone has speed \(v\) and it experiences a resistance force of magnitude \(\lambda mv\), where \(\lambda\) is a constant.

1. Show that $$\frac{\text{d}v}{\text{d}t} = g - \lambda v$$ [2 marks]
2. The initial speed of the stone is \(u\). Find an expression for \(v\) at time \(t\). [6 marks]

A particle \(P\) of mass \(0.5\) kg is released from rest at time \(t = 0\) and falls vertically through a liquid. The motion of \(P\) is resisted by a force of magnitude \(2v\) N, where \(v\) m s\(^{-1}\) is the speed of \(v\) at time \(t\) seconds.

1. Show that \(5 \frac{\mathrm{d}v}{\mathrm{d}t} = 49 - 20v\). [2]
2. Find the speed of \(P\) when \(t = 1\). [5]

A particle \(P\) of mass \(0.5\) kg falls vertically from rest. After \(t\) seconds it has speed \(v\) m s\(^{-1}\). A resisting force of magnitude \(1.5v\) newtons acts on \(P\) as it falls.

1. Show that \(3v = 9.8(1 - e^{-3t})\). [8]
2. Find the distance that \(P\) falls in the first two seconds of its motion. [5]

A small ball of mass \(m\) is projected vertically upwards from a point \(O\) with speed \(U\). The ball is subject to air resistance of magnitude \(mkv\), where \(v\) is the speed of the ball and \(k\) is a positive constant. Find, in terms of \(U\), \(g\) and \(k\), the maximum height above \(O\) reached by the ball. (8)

A stone of mass \(m\) is projected vertically upwards with initial velocity \(u\). At time \(t\), the height risen above the point of projection is \(x\) and the resistance to motion is \(kv\) when the velocity of the stone is \(v\).

1. Write down a first-order differential equation relating \(v\) and \(t\) and hence find \(t\) in terms of \(v\). [5]
2. Write down a first-order differential equation relating \(v\) and \(x\) and hence find \(x\) in terms of \(v\). [6]