Velocity from acceleration by integration

3 A particle moves in a straight line starting from rest from a point \(O\). The acceleration of the particle at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 4 t ^ { \frac { 1 } { 2 } }\).

1. Find the speed of the particle when \(t = 9\).
2. Find the time after leaving \(O\) at which the speed (in metres per second) and the distance travelled (in metres) are numerically equal.

7 A particle \(P\) moves in a straight line, starting from a point \(O\) with velocity \(1.72 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the particle, \(t \mathrm {~s}\) after leaving \(O\), is given by \(a = 0.1 t ^ { \frac { 3 } { 2 } }\).

1. Find the value of \(t\) when the velocity of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the displacement of \(P\) from \(O\) when \(t = 2\), giving your answer correct to 2 decimal places. [3]

6 A particle \(P\) of mass 0.6 kg is projected vertically upwards with speed \(5.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) which is 6.2 m above the ground. Air resistance acts on \(P\) so that its deceleration is \(10.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when \(P\) is moving upwards, and its acceleration is \(9.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when \(P\) is moving downwards. Find

1. the greatest height above the ground reached by \(P\),
2. the speed with which \(P\) reaches the ground,
3. the total work done on \(P\) by the air resistance.

3 A particle \(P\) moves in a straight line, starting from the point \(O\) with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The acceleration of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is \(2 t ^ { \frac { 2 } { 3 } } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Show that \(t ^ { \frac { 5 } { 3 } } = \frac { 5 } { 6 }\) when the velocity of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the distance of \(P\) from \(O\) when the velocity of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

4 An aeroplane moves along a straight horizontal runway before taking off. It starts from rest at \(O\) and has speed \(90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the instant it takes off. While the aeroplane is on the runway at time \(t\) seconds after leaving \(O\), its acceleration is \(( 1.5 + 0.012 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Find

1. the value of \(t\) at the instant the aeroplane takes off,
2. the distance travelled by the aeroplane on the runway.

8. \begin{figure}[h] \end{figure} The acceleration-time graph shown in Figure 5 represents part of a journey made by a car along a straight horizontal road. The car accelerated from rest at time \(t = 0\)

1. Find the distance travelled by the car during the first 4 s of its journey.
2. Find the total distance travelled by the car during the first 26s of its journey.
   

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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)

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 \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) ?

2 A particle, of mass 50 kg , moves on a smooth horizontal plane. A single horizontal force $$\left[ \left( 300 t - 60 t ^ { 2 } \right) \mathbf { i } + 100 \mathrm { e } ^ { - 2 t } \mathbf { j } \right] \text { newtons }$$ acts on the particle at time \(t\) seconds.
The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.

1. Find the acceleration of the particle at time \(t\).
2. When \(t = 0\), the velocity of the particle is \(( 7 \mathbf { i } - 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Find the velocity of the particle at time \(t\).
3. Calculate the speed of the particle when \(t = 1\).

3 A particle, of mass 10 kg , moves on a smooth horizontal plane. At time \(t\) seconds, the acceleration of the particle is given by $$\left\{ \left( 40 t + 3 t ^ { 2 } \right) \mathbf { i } + 20 \mathrm { e } ^ { - 4 t } \mathbf { j } \right\} \mathrm { m } \mathrm {~s} ^ { - 2 }$$ where the vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.

1. At time \(t = 1\), the velocity of the particle is \(\left( 6 \mathbf { i } - 5 \mathrm { e } ^ { - 4 } \mathbf { j } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find the velocity of the particle at time \(t\).
2. Calculate the initial speed of the particle.

4. The acceleration \(a \mathrm {~ms} ^ { - 2 }\) of a particle \(P\) moving in a straight line away from a fixed point \(O\) is given by \(a = \frac { k } { 1 + t }\), where \(t \mathrm {~s}\) is the time that has elapsed since \(P\) left \(O\), and \(k\) is a constant.

1. By solving a suitable differential equation, find an expression for the velocity \(v \mathrm {~ms} ^ { - 1 }\) of \(P\) in terms of \(t , k\) and another constant \(c\). Given that \(v = 0\) when \(t = 0\) and that \(v = 4\) when \(t = 2\),
2. show that \(v \ln 3 = 4 \ln ( 1 + t )\).
3. Calculate the time when \(P\) has a speed of \(8 \mathrm {~ms} ^ { - 1 }\). \section*{MECHANICS 3 (A)TEST PAPER 4 Page 2}

5 At time \(t\) seconds, where \(t \geqslant 0\), 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 \(( 2 \cos 2 t \mathbf { i } + 4 \sin t \mathbf { j } ) \mathrm { N }\). The acceleration of P at time t seconds is \(\mathrm { am } \mathrm { S } ^ { - 2 }\).

1. Find a in terms of t. The particle P is at rest when \(\mathrm { t } = 0\).
2. Determine the speed of P at the instant when \(\mathrm { t } = 2\). Answer all the questions.
   Section B (91 marks)

2 A particle \(P\) moves with acceleration \(( - 3 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Initially the velocity of \(P\) is \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the velocity of \(P\) at time \(t\) seconds.
2. Find the speed of \(P\) when \(t = 0.5\).

8 A boat is initially at the origin, heading due east at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then experiences a constant acceleration of \(( - 0.2 \mathbf { i } + 0.25 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. State the initial velocity of the boat as a vector.
2. Find an expression for the velocity of the boat \(t\) seconds after it has started to accelerate.
3. Find the value of \(t\) when the boat is travelling due north.
4. Find the bearing of the boat from the origin when the boat is travelling due north.

13 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively.
At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) of mass 2 kg is moving on a smooth horizontal surface under the action of a constant horizontal force \(( - 8 \mathbf { i } - 54 \mathbf { j } ) \mathrm { N }\) and a variable horizontal force \(\left( 4 t \mathbf { i } + 6 ( 2 t - 1 ) ^ { 2 } \mathbf { j } \right) \mathrm { N }\).

1. Determine the value of \(t\) when the forces acting on \(P\) are in equilibrium. It is given that \(P\) is at rest when \(t = 0\).
2. Determine the speed of \(P\) at the instant when \(P\) is moving due north.
3. Determine the distance between the positions of \(P\) when \(t = 0\) and \(t = 3\).

15 A particle, \(P\), is moving in a straight line with acceleration \(a \mathrm {~ms} ^ { - 2 }\) at time \(t\) seconds, where $$a = 4 - 3 t ^ { 2 }$$ 15

1. Initially \(P\) is stationary.
   Find an expression for the velocity of \(P\) in terms of \(t\).
   
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8 A particle of mass 4 kg moves horizontally in a straight line. At time \(t\) seconds the velocity of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
The following horizontal forces act on the particle:

• a constant driving force of magnitude 1.8 newtons
• another driving force of magnitude \(30 \sqrt { t }\) newtons
• a resistive force of magnitude \(0.08 v ^ { 2 }\) newtons

When \(t = 70 , v = 54\)
Use Euler's method with a step length of 0.5 to estimate the velocity of the particle after 71 seconds. Give your answer to four significant figures.