Acceleration as function of velocity (separation of variables)

3. A particle \(P\) is moving in a straight line. At time \(t\) seconds, \(P\) is at a distance \(x\) metres from a fixed point \(O\) on the line and is moving away from \(O\) with speed \(\frac { 10 } { x + 6 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the acceleration of \(P\) when \(x = 14\) Given that \(x = 2\) when \(t = 1\),
2. find the value of \(t\) when \(x = 14\)

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)

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]

3. A body, of mass 9 kg , is projected along a straight horizontal track with an initial speed of \(20 \mathrm {~ms} ^ { - 1 }\). At time \(t \mathrm {~s}\) the body experiences a resistance of magnitude \(( 0.2 + 0.03 v ) \mathrm { N }\) where \(v \mathrm {~ms} ^ { - 1 }\) is its speed.

1. Show that \(v\) satisfies the differential equation $$900 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - ( 20 + 3 v )$$
2. Find an expression for \(t\) in terms of \(v\).
3. Calculate, to the nearest second, the time taken for the body to come to rest.

1. A particle, \(P\), moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) in the direction of \(x\) increasing and the acceleration of \(P\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) in the direction of \(x\) increasing.

When \(t = 0\) the particle is at rest at the origin \(O\).
Given that \(a = \frac { 5 } { 2 } ( 5 - v )\)

1. show that \(v = 5 \left( 1 - \mathrm { e } ^ { - 2.5 t } \right)\)
2. state the limiting value of \(v\) as \(t\) increases. At the instant when \(v = 2.5\), the particle is \(d\) metres from \(O\).
3. Show that \(d = 2 \ln 2 - 1\)

1. A car moves in a straight line along a horizontal road. The car is modelled as a particle. At time \(t\) seconds, where \(t \geqslant 0\), the speed of the car is \(v \mathrm {~ms} ^ { - 1 }\)

At the instant when \(t = 0\), the car passes through the point \(A\) with speed \(2 \mathrm {~ms} ^ { - 1 }\) The acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), of the car is modelled by $$a = \frac { 4 } { 2 + v }$$ in the direction of motion of the car.

1. Use algebraic integration to show that \(v = \sqrt { 8 t + 16 } - 2\) At the instant when the car passes through the point \(B\), the speed of the car is \(4 \mathrm {~ms} ^ { - 1 }\)
2. Use algebraic integration to find the distance \(A B\).

1. At time \(t = 0\), a toy electric car is at rest at a fixed point \(O\). The car then moves in a horizontal straight line so that at time \(t\) seconds \(( t > 0 )\) after leaving \(O\), the velocity of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the acceleration of the car is modelled as \(( p + q v ) \mathrm { ms } ^ { - 2 }\), where \(p\) and \(q\) are constants.

When \(t = 0\), the acceleration of the car is \(3 \mathrm {~ms} ^ { - 2 }\) When \(t = T\), the acceleration of the car is \(\frac { 1 } { 2 } \mathrm {~ms} ^ { - 2 }\) and \(v = 4\)

1. Show that $$8 \frac { \mathrm {~d} v } { \mathrm {~d} t } = ( 24 - 5 v )$$
2. Find the exact value of \(T\), simplifying your answer.

1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\) in the direction of \(x\) increasing, where

$$v = \frac { 1 } { 2 } \left( 3 \mathrm { e } ^ { 2 t } - 1 \right) \quad t \geqslant 0$$ The acceleration of \(P\) at time \(t\) seconds is \(a \mathrm {~ms} ^ { - 2 }\)

1. Show that \(a = 2 v + 1\)
2. Find the acceleration of \(P\) when \(t = 0\)
3. Find the exact distance travelled by \(P\) in accelerating from a speed of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. A particle \(P\) is moving along the \(x\)-axis. At time \(t\) seconds, \(P\) has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction and acceleration \(a \mathrm {~ms} ^ { - 2 }\) in the positive \(x\) direction.

In a model of the motion of \(P\) $$a = 4 - 3 v$$ When \(t = 0 , v = 0\)

1. Use integration to show that \(v = k \left( 1 - \mathrm { e } ^ { - 3 t } \right)\), where \(k\) is a constant to be found. When \(t = 0 , P\) is at the origin \(O\)
2. Find, in terms of \(t\) only, the distance of \(P\) from \(O\) at time \(t\) seconds.

5 A car, of mass 1600 kg , is travelling along a straight horizontal road at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the driving force is removed. The car then freewheels and experiences a resistance force. The resistance force has magnitude \(40 v\) newtons, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car after it has been freewheeling for \(t\) seconds. Find an expression for \(v\) in terms of \(t\).

8 A stone, of mass 0.05 kg , is moving along the smooth horizontal floor of a tank, which is filled with oil. At time \(t\), the stone has speed \(v\). As the stone moves, it experiences a resistance force of magnitude \(0.08 v ^ { 2 }\).

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 1.6 v ^ { 2 }$$ (2 marks)
2. The initial speed of the stone is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Show that $$v = \frac { 15 } { 5 + 24 t }$$ (5 marks)

17 A ball is released from a great height so that it falls vertically downwards towards the surface of the Earth. 17

1. Using a simple model, Andy predicts that the velocity of the ball, exactly 2 seconds after being released from rest, is \(2 g \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Show how Andy has obtained his prediction.
   17
2. Using a refined model, Amy predicts that the ball's acceleration, \(a \mathrm {~ms} ^ { - 2 }\), at time \(t\) seconds after being released from rest is $$a = g - 0.1 v$$ where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the ball at time \(t\) seconds. Find an expression for \(v\) in terms of \(t\).
   17
3. Comment on the value of \(v\) for the two models as \(t\) becomes large.

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

1. Show that \(\frac{9}{15-v} \frac{dv}{dt} = 1\). [2]
2. Solve this differential equation to show that \(v = 15(1-e^{-\frac{t}{9}})\). [4]
3. Find the distance travelled by the cyclist in the first \(9 \text{ s}\) of the motion. [4]

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

1. Show that \(\frac{dv}{dt} = 10 - 0.2v\). Hence find \(v\) in terms of \(t\). [6]
2. Given that the ball reaches the ground when \(t = 2\), calculate \(h\). [4]

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

1. Show that the acceleration of \(P\) is \((10 - 4v)\) m s\(^{-2}\). [2]
2. Find the value of \(v\) when \(t = 0.6\). [5]

A particle \(P\) is moving along a straight line with acceleration \(3ku - kv\) where \(v\) is its velocity at time \(t\), \(u\) is its initial velocity and \(k\) is a constant. The velocity and acceleration of \(P\) are both in the direction of increasing displacement from the initial position.

1. Find the time taken for \(P\) to achieve a velocity of \(2u\). [3]
2. Find an expression for the displacement of \(P\) from its initial position when its velocity is \(2u\). [5]

A particle \(Q\) of mass \(m\) kg falls from rest under gravity. The motion of \(Q\) is resisted by a force of magnitude \(mkv\) N, where \(v\) ms\(^{-1}\) is the speed of \(Q\) at time \(t\) s and \(k\) is a positive constant. Find an expression for \(v\) in terms of \(g\), \(k\) and \(t\). [6]

A particle \(P\) of mass 1 kg is moving along a straight line against a resistive force of magnitude \(\frac{10\sqrt{v}}{(t+1)^2}\) N, where \(v\) ms\(^{-1}\) is the speed of \(P\) at time \(t\)s. When \(t = 0\), \(v = 25\). Find an expression for \(v\) in terms of \(t\). [5]

A parachutist of mass \(m\) kg opens his parachute when he is moving vertically downwards with a speed of \(50\text{ ms}^{-1}\). At time \(t\) s after opening his parachute, he has fallen a distance \(x\) m from the point where he opened his parachute, and his speed is \(v\text{ ms}^{-1}\). The forces acting on him are his weight and a resistive force of magnitude \(mv\) N.

1. Find an expression for \(v\) in terms of \(t\). [6]
2. Find an expression for \(x\) in terms of \(t\). [3]
3. Find the distance that the parachutist has fallen, since opening his parachute, when his speed is \(15\text{ ms}^{-1}\). [2]

A particle \(P\) of mass \(m\) kg moves along a horizontal straight line with acceleration \(a\) ms\(^{-2}\) given by $$a = \frac{v(1-2t^2)}{t},$$ where \(v\) ms\(^{-1}\) is the velocity of \(P\) at time \(t\) s.

1. Find an expression for \(v\) in terms of \(t\) and an arbitrary constant. [3]
2. Given that \(a = 5\) when \(t = 1\), find an expression, in terms of \(m\) and \(t\), for the horizontal force acting on \(P\) at time \(t\). [3]

A particle \(P\) of mass 2 kg moves in a straight line along a smooth horizontal plane. The only horizontal force acting on \(P\) is a resistance of magnitude \(4v\) N, where \(v\) m s\(^{-1}\) is its speed. At time \(t = 0\) s, \(P\) has a speed of 5 m s\(^{-1}\). Find \(v\) in terms of \(t\). [6]