Velocity from acceleration by integration

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.

1 A particle travels along a straight line. Its acceleration during the time interval \(0 \leqslant t \leqslant 8\) is given by the acceleration-time graph in Fig. 1. \begin{figure}[h] \end{figure}

1. Write down the acceleration of the particle when \(t = 4\). Given that the particle starts from rest, find its speed when \(t = 4\).
2. Write down an expression in terms of \(t\) for the acceleration, \(a \mathrm {~ms} ^ { - 2 }\), of the particle in the time interval \(0 \leqslant t \leqslant 4\).
3. Without calculation, state the time at which the speed of the particle is greatest. Give a reason for your answer.
4. Calculate the change in speed of the particle from \(t = 5\) to \(t = 8\), indicating whether this is an increase or a decrease.

1 A particle P moves on the x-axis. The acceleration of P at time t seconds, \(\mathrm { t } \geqslant 0\), is \(( 3 \mathrm { t } + 5 ) \mathrm { ms } ^ { - 2 }\) in the positive x -direction. When \(\mathrm { t } = 0\), the velocity of P is \(2 \mathrm {~ms} ^ { - 1 }\) in the positive x -direction. When \(\mathrm { t } = \mathrm { T }\), the velocity of P is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive x -direction. Find the value of T .
(6)

6. At time \(t\) seconds the acceleration, a \(\mathrm { m } \mathrm { s } ^ { - 2 }\), of a particle \(P\) relative to a fixed origin \(O\), is given by \(\mathbf { a } = 2 \mathbf { i } + 6 t \mathbf { j }\). Initially the velocity of \(P\) is \(( 2 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the velocity of \(P\) at time \(t\) seconds. At time \(t = 2\) seconds the particle \(P\) is given an impulse ( \(3 \mathbf { i } - 1.5 \mathbf { j }\) ) Ns. Given that the particle \(P\) has mass 0.5 kg ,
2. find the speed of \(P\) immediately after the impulse has been applied.

2. A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its acceleration is \(2 \sin \frac { 1 } { 2 } t \mathrm {~m} \mathrm {~s} ^ { - 2 }\), both measured in the direction of \(O x\). Given that \(v = 4\) when \(t = 0\),

1. find \(v\) in terms of \(t\),
2. calculate the distance travelled by \(P\) between the times \(t = 0\) and \(t = \frac { \pi } { 2 }\).

3 A car is travelling along a straight horizontal road with velocity \(32.5 \mathrm {~ms} ^ { - 1 }\). The driver applies the brakes and the car decelerates at \(( 8 - 0.6 t ) \mathrm { ms } ^ { - 2 }\), where \(t \mathrm {~s}\) is the time which has elapsed since the brakes were first applied.

1. Show that, while the car is decelerating, its velocity is \(\left( 32.5 - 8 t + 0.3 t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Find the time taken to bring the car to rest.
3. Show that the distance travelled while the car is decelerating is 75 m .

4 The acceleration of a particle \(P\) moving in a straight line is \(\left( t ^ { 2 } - 9 t + 18 \right) \mathrm { ms } ^ { - 2 }\), where \(t\) is the time in seconds.

1. Find the values of \(t\) for which the acceleration is zero.
2. It is given that when \(t = 3\) the velocity of \(P\) is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the velocity of \(P\) when \(t = 0\).
3. Show that the direction of motion of \(P\) changes before \(t = 1\).

2 Fig. 2 shows an acceleration-time graph modelling the motion of a particle. \begin{figure}[h] \end{figure} At \(t = 0\) the particle has a velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive direction.

1. Find the velocity of the particle when \(t = 2\).
2. At what time is the particle travelling in the negative direction with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) ?

9 A particle is moving in a straight line. The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the particle at time \(t \mathrm {~s}\) is given by \(\mathrm { a } = 0.8 \mathrm { t } + 0.5\). The initial velocity of the particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Determine whether the particle is ever stationary.

2 Fig. 2 shows an acceleration-time graph modelling the motion of a particle. \begin{figure}[h] \end{figure} At \(t = 0\) the particle has a velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive direction.

1. Find the velocity of the particle when \(t = 2\).
2. At what time is the particle travelling in the negative direction with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) ?

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

A particle \(P\) moves along a straight line. Initially \(P\) is at rest at the point \(O\) on the line. At time \(t\) seconds, where \(t \geqslant 0\)

• the displacement of \(P\) from \(O\) is \(x\) metres
• the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction
• the acceleration of \(P\) is \(\frac { 96 } { ( 3 t + 5 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) in the positive \(x\) direction
  1. Show that, at time \(t\) seconds, \(v = p - \frac { q } { ( 3 t + 5 ) ^ { 2 } }\), where \(p\) and \(q\) are constants to be determined.
  2. Find the limiting value of \(v\) as \(t\) increases.
  3. Find the value of \(x\) when \(t = 2\)

A particle moves in a straight line. At time \(t\) seconds its velocity is \(v\) ms\(^{-1}\) and its acceleration is \(a\) ms\(^{-2}\).

1. Given that \(a = —\), express \(v\) in terms of \(t\).
2. Given that \(v = tv\) when \(t = 0\), find \(v\) in terms of \(t\).
3. Find the displacement from the starting point when \(t = v\).

[6]

A particle \(P\) moves in a straight line and passes through a point \(O\) of the line with velocity \(2\text{ m s}^{-1}\). At time \(t\) s after passing through \(O\), the velocity of \(P\) is \(v\text{ m s}^{-1}\) and the acceleration of \(P\) is given by \(\text{e}^{-0.5t}\text{ m s}^{-2}\). Calculate the velocity of \(P\) when \(t = 1.2\). [4]

A particle \(P\) moves in a straight line and passes through a point \(O\) of the line with velocity \(2\,\text{m s}^{-1}\). At time \(t\) s after passing through \(O\), the velocity of \(P\) is \(v\,\text{m s}^{-1}\) and the acceleration of \(P\) is given by \(e^{-0.5t}\,\text{m s}^{-2}\). Calculate the velocity of \(P\) when \(t = 1.2\). [4]

A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(\mathbf{F}\) newtons. At time \(t\) seconds, \(\mathbf{F} = (1.5t^2 - 3)\mathbf{i} + 2t\mathbf{j}\). When \(t = 2\), the velocity of \(P\) is \((-4\mathbf{i} + 5\mathbf{j})\) m s\(^{-1}\).

1. Find the acceleration of \(P\) at time \(t\) seconds. [2]
2. Show that, when \(t = 3\), the velocity of \(P\) is \((9\mathbf{i} + 15\mathbf{j})\) m s\(^{-1}\). [5]

When \(t = 3\), the particle \(P\) receives an impulse \(\mathbf{Q}\) N s. Immediately after the impulse the velocity of \(P\) is \((-3\mathbf{i} + 20\mathbf{j})\) m s\(^{-1}\). Find

1. the magnitude of \(\mathbf{Q}\), [3]
2. the angle between \(\mathbf{Q}\) and \(\mathbf{i}\). [3]

A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, its acceleration is \((5 - 2t)\) m s\(^{-2}\), measured in the direction of \(x\) increasing. When \(t = 0\), its velocity is 6 m s\(^{-1}\) measured in the direction of \(x\) increasing. Find the time when \(P\) is instantaneously at rest in the subsequent motion. [6]

A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\) at time \(t\) seconds, \(t \geq 0\), is \((3t + 5)\) m s\(^{-2}\) in the positive \(x\)-direction. When \(t = 0\), the velocity of \(P\) is 2 m s\(^{-1}\) in the positive \(x\)-direction. When \(t = T\), the velocity of \(P\) is 6 m s\(^{-1}\) in the positive \(x\)-direction. Find the value of \(T\). (6)

A particle \(P\) moves along the x-axis in the positive direction. At time \(t\) seconds, the velocity of \(P\) is \(v\) m s\(^{-1}\) and its acceleration is \(\frac{1}{5}e^{-2t}\) m s\(^{-2}\). When \(t = 0\) the speed of \(P\) is 10 m s\(^{-1}\).

1. Express \(v\) in terms of \(t\). [4]
2. Find, to 3 significant figures, the speed of \(P\) when \(t = 3\). [2]
3. Find the limiting value of \(v\). [1]

A particle \(P\), initially at rest at the point \(O\), moves in a straight line such that at time \(t\) seconds after leaving \(O\) its acceleration is \((12t - 15)\) ms\(^{-2}\). Find

1. the velocity of \(P\) at time \(t\) seconds after it leaves \(O\), [3 marks]
2. the value of \(t\) when the speed of \(P\) is 36 ms\(^{-1}\). [3 marks]

The acceleration of a particle \(P\) is \((8t - 18)\) ms\(^{-2}\), where \(t\) seconds is the time that has elapsed since \(P\) passed through a fixed point \(O\) on the straight line on which it is moving. At time \(t = 3\), \(P\) has speed \(2\) ms\(^{-1}\). Find

1. the velocity of \(P\) at time \(t\), [4 marks]
2. the values of \(t\) when \(P\) is instantaneously at rest. [3 marks]

A particle \(Q\) of mass \(0.2\) kg is projected horizontally with velocity \(4\) m s\(^{-1}\) from a fixed point \(A\) on a smooth horizontal surface. At time \(t\) s after projection \(Q\) is \(x\) m from \(A\) and is moving away from \(A\) with velocity \(v\) m s\(^{-1}\). There is a force of \(3\cos 2t\) N acting on \(Q\) in the positive \(x\)-direction.

1. Find an expression for the velocity of \(Q\) at time \(t\). State the maximum and minimum values of the velocity of \(Q\) as \(t\) varies. [4]
2. Find the average velocity of \(Q\) between times \(t = \pi\) and \(t = \frac{3}{2}\pi\). [4]

A particle, P, is moving along a straight line such that its acceleration \(a\) m s⁻², at any time, \(t\) seconds, may be modelled by $$a = 3 + 0.2t$$ When \(t = 2\), the velocity of P is \(k\) m s⁻¹

1. Show that the initial velocity of P is given by the expression \((k - 6.4)\) m s⁻¹ [4 marks]
2. The initial velocity of P is one fifth of the velocity when \(t = 2\) Find the value of \(k\). [2 marks]

At time \(t\) seconds, where \(t \geq 0\), a particle P of mass 2 kg is moving on a smooth horizontal surface. The particle moves under the action of a constant horizontal force of \((-2\mathbf{i} + 6\mathbf{j})\) N and a variable horizontal force of \((2\cos 2t \mathbf{i} + 4\sin t \mathbf{j})\) N. The acceleration of P at time \(t\) seconds is \(\mathbf{a}\) m s\(^{-2}\).

1. Find \(\mathbf{a}\) in terms of \(t\). [2]

The particle P is at rest when \(t = 0\).

1. Determine the speed of P at the instant when \(t = 2\). [5]

A particle, of mass 4 kg, moves in a straight line under the action of a single force \(F\) N, whose magnitude at time \(t\) seconds is given by $$F = 12\sqrt{t} - 32 \quad \text{for} \quad t \geqslant 0.$$

1. Find the acceleration of the particle when \(t = 9\). [2]
2. Given that the particle has velocity \(-1\text{ms}^{-1}\) when \(t = 4\), find an expression for the velocity of the particle at \(t\) s. [3]
3. Determine whether the speed of the particle is increasing or decreasing when \(t = 9\). [2]