3.02f Non-uniform acceleration: using differentiation and integration

10 A particle \(P\) is moving in a straight line. At time \(t\) seconds \(P\) has velocity \(v \mathrm {~ms} ^ { - 1 }\) where \(v = ( 2 t + 1 ) ( 3 - t )\).

1. Find the deceleration of \(P\) when \(t = 4\).
2. State the positive value of \(t\) for which \(P\) is instantaneously at rest.
3. Find the total distance that \(P\) travels between times \(t = 0\) and \(t = 4\).

12 \includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-09_647_935_260_242} A particle \(P\) moves in a straight line. At time \(t\) seconds, where \(t \geqslant 0\), the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It is given that \(v = - 3 t ^ { 2 } + 24 t + k\), where \(k\) is a positive constant. The diagram shows the velocity-time graph for the motion of \(P\). \(P\) attains its maximum velocity at time \(T\) seconds. Given that the distance travelled by \(P\) between times \(t = 1\) and \(t = T\) is 297 m , determine the time when \(P\) is instantaneously at rest. \section*{END OF QUESTION PAPER}

11 \includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-08_586_672_1231_242} A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds, where \(t \geqslant 0\), the velocity of \(P\) in the positive \(x\)-direction is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It is given that \(v = t ( t - 3 ) ( 8 - t )\). \(P\) attains its maximum velocity at time \(T\) seconds. The diagram shows part of the velocity-time graph for the motion of \(P\).

1. State the acceleration of \(P\) at time \(T\).
2. In this question you must show detailed reasoning. Determine the value of \(T\).
3. Find the total distance that \(P\) travels between times \(t = 0\) and \(t = T\). \includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-09_524_410_251_242} Particles \(P\) and \(Q\), of masses 4 kg and 6 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is in equilibrium with \(P\) hanging 1.75 m above a horizontal plane and \(Q\) resting on the plane. Both parts of the string below the pulley are vertical (see diagram).
   1. Find the magnitude of the normal reaction force acting on \(Q\). The mass of \(P\) is doubled, and the system is released from rest. You may assume that in the subsequent motion \(Q\) does not reach the pulley.
   2. Determine the magnitude of the force exerted on the pulley by the string before \(P\) strikes the plane.
   3. Determine the total distance travelled by \(Q\) between the instant when the system is released and the instant when \(Q\) first comes momentarily to rest. When this motion is observed in practice, it is found that the total distance travelled by \(Q\) between the instant when the system is released and the instant when \(Q\) first comes momentarily to rest is less than the answer calculated in part (c).
   4. State one factor that could account for this difference.

10 A particle \(P\) is moving in a straight line. At time \(t\) seconds, where \(t \geqslant 0 , P\) has velocity \(v \mathrm {~ms} ^ { - 1 }\) and acceleration \(a \mathrm {~ms} ^ { - 2 }\) where \(a = 4 t - 9\). It is given that \(v = 2\) when \(t = 1\).

1. Find an expression for \(v\) in terms of \(t\). The particle \(P\) is instantaneously at rest when \(t = t _ { 1 }\) and \(t = t _ { 2 }\), where \(t _ { 1 } < t _ { 2 }\).
2. Find the values of \(t _ { 1 }\) and \(t _ { 2 }\).
3. Determine the total distance travelled by \(P\) between times \(t = 0\) and \(t = t _ { 2 }\).

7 The velocity of a particle moving in a straight line is modelled by \(\mathbf { v } = 0.6 \mathbf { t } ^ { 2 } - 2.1 \mathbf { t } + 1.5\) where \(v\) is the velocity in metres per second and \(t\) is the time in seconds.

1. Determine the times at which the particle is stationary.
2. Find the acceleration of the particle at the first of the times at which it is stationary.
3. Find the distance travelled by the particle between the times at which it is stationary.

9 Two trains are travelling in the same direction on parallel straight tracks and train A overtakes train B . At time \(t\) seconds after the front of train A overtakes the front of train B the velocities of trains A and B are \(v _ { \mathrm { A } } \mathrm { m } \mathrm { s } ^ { - 1 }\) and \(v _ { \mathrm { B } } \mathrm { ms } ^ { - 1 }\) respectively. The velocity of train A is modelled by \(\mathrm { v } _ { \mathrm { A } } = 25 - 0.6 \mathrm { t }\). The velocity-time graph of train A is shown below. \includegraphics[max width=\textwidth, alt={}, center]{b5c47a93-ce43-4aa1-ba7f-fbb650523373-5_664_1399_550_242}

1. A student argues that the speed of train A changes by \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 30 seconds so its acceleration is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Comment on the validity of the student's argument.
2. When the front of train A overtakes the front of train B , train B has a velocity of \(10 \mathrm {~ms} ^ { - 1 }\). The acceleration of train \(B\) is constant and is modelled as \(0.15 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Write down the equation for \(v _ { \mathrm { B } }\) in terms of \(t\) that models the velocity of train B .
3. Draw the velocity-time graph of train B on the copy of the diagram in the Printed Answer Booklet.
4. Determine the distance between the fronts of the trains at the time when the trains are travelling at the same velocity.
5. Explain why the model for train A would not be valid for large values of \(t\).

6 In this question you must show detailed reasoning.
A particle moves in a straight line. Its velocity \(v \mathrm {~ms} ^ { - 1 }\) after \(t \mathrm {~s}\) is given by \(\mathrm { v } = \mathrm { t } ^ { 3 } - 5 \mathrm { t } ^ { 2 }\).

1. Find the times at which the particle is stationary.
2. Find the total distance travelled by the particle in the first 6 seconds.

5 The position vector \(\mathbf { r }\) metres of a particle at time \(t\) seconds is given by $$\mathbf { r } = \left( 1 + 12 t - 2 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 2 } - 6 t \right) \mathbf { j }$$

1. Find an expression for the velocity of the particle at time \(t\).
2. Determine whether the particle is ever stationary.

12 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the \(x\) - and \(y\)-directions respectively.
The velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of a particle is given by \(\mathbf { v } = 3 \mathbf { i } + \left( 6 t ^ { 2 } - 5 \right) \mathbf { j }\). The initial position of the particle is \(7 \mathbf { j } \mathrm {~m}\).

1. Find an expression for the position vector of the particle at time \(t \mathrm {~s}\).
2. Find the Cartesian equation of the path of the particle.

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 A box of emergency supplies is dropped to victims of a natural disaster from a stationary helicopter at a height of 1000 metres. The initial velocity of the box is zero. At time \(t \mathrm {~s}\) after being dropped, the acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), of the box in the vertically downwards direction is modelled by $$\begin{aligned} & a = 10 - t \text { for } 0 \leqslant t \leqslant 10 \\ & a = 0 \quad \text { for } \quad t > 10 \end{aligned}$$

1. Find an expression for the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the box in the vertically downwards direction in terms of \(t\) for \(0 \leqslant t \leqslant 10\). Show that for \(t > 10 , v = 50\).
2. Draw a sketch graph of \(v\) against \(t\) for \(0 \leqslant t \leqslant 20\).
3. Show that the height, \(h \mathrm {~m}\), of the box above the ground at time \(t\) s is given, for \(0 \leqslant t \leqslant 10\), by $$h = 1000 - 5 t ^ { 2 } + \frac { 1 } { 6 } t ^ { 3 }$$ Find the height of the box when \(t = 10\).
4. Find the value of \(t\) when the box hits the ground.
5. Some of the supplies in the box are damaged when the box hits the ground. So measures are considered to reduce the speed with which the box hits the ground the next time one is dropped. Two different proposals are made. Carry out suitable calculations and then comment on each of them.
   (A) The box should be dropped from a height of 500 m instead of 1000 m .
   (B) The box should be fitted with a parachute so that its acceleration is given by $$\begin{gathered} \quad a = 10 - 2 t \text { for } 0 \leqslant t \leqslant 5 , \\ a = 0 \quad \text { for } \quad t > 5 . \end{gathered}$$

4 A particle moves along a straight line through an origin O . Initially the particle is at O .
At time \(t \mathrm {~s}\), its displacement from O is \(x \mathrm {~m}\) and its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$v = 24 - 18 t + 3 t ^ { 2 }$$

1. Find the times, \(T _ { 1 } \mathrm {~s}\) and \(T _ { 2 } \mathrm {~s}\) (where \(T _ { 2 } > T _ { 1 }\) ), at which the particle is stationary.
2. Find an expression for \(x\) at time \(t\) s. Show that when \(t = T _ { 1 } , x = 20\) and find the value of \(x\) when \(t = T _ { 2 }\).

2 \begin{figure}[h] \end{figure} A toy car is moving along the straight line \(\mathrm { O } x\), where O is the origin. The time \(t\) is in seconds. At time \(t = 0\) the car is at \(\mathrm { A } , 3 \mathrm {~m}\) from O as shown in Fig. 5. The velocity of the car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$v = 2 + 12 t - 3 t ^ { 2 }$$ Calculate the distance of the car from O when its acceleration is zero.

3 A particle moves along a straight line containing a point O . Its displacement, \(x \mathrm {~m}\), from O at time \(t\) seconds is given by $$x = 12 t - t ^ { 3 } , \text { where } - 10 \leqslant t \leqslant 10$$ Find the values of \(x\) for which the velocity of the particle is zero.

4 A point P on a piece of machinery is moving in a vertical straight line. The displacement of P above ground level at time \(t\) seconds is \(y\) metres. The displacement-time graph for the motion during the time interval \(0 \leqslant t \leqslant 4\) is shown in Fig. 7 . \begin{figure}[h] \end{figure}

1. Using the graph, determine for the time interval \(0 \leqslant t \leqslant 4\) (A) the greatest displacement of P above its position when \(t = 0\),
   (B) the greatest distance of P from its position when \(t = 0\),
   (C) the time interval in which P is moving downwards,
   (D) the times when P is instantaneously at rest. The displacement of P in the time interval \(0 \leqslant t \leqslant 3\) is given by \(y = - 4 t ^ { 2 } + 8 t + 12\).
2. Use calculus to find expressions in terms of \(t\) for the velocity and for the acceleration of P in the interval \(0 \leqslant t \leqslant 3\).
3. At what times does P have a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the interval \(0 \leqslant t \leqslant 3\) ? In the time interval \(3 \leqslant t \leqslant 4 , \mathrm { P }\) has a constant acceleration of \(32 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is no sudden change in velocity when \(t = 3\).
4. Find an expression in terms of \(t\) for the displacement of P in the interval \(3 \leqslant t \leqslant 4\).

3 A particle is moving along a straight line and its position is relative to an origin on the line. At time \(t \mathrm {~s}\), the particle's acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), is given by $$a = 6 t - 12 .$$ At \(t = 0\) the velocity of the particle is \(+ 9 \mathrm {~ms} ^ { - 1 }\) and its position is - 2 m .

1. Find an expression for the velocity of the particle at time \(t \mathrm {~s}\) and verify that it is stationary when \(t = 3\).
2. Find the position of the particle when \(t = 2\).

5 A particle is moving along a straight line and its position is relative to an origin on the line. At time \(t \mathrm {~s}\), the particle's acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), is given by $$a = 6 t - 12 .$$ At \(t = 0\) the velocity of the particle is \(+ 9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its position is - 2 m .

1. Find an expression for the velocity of the particle at time \(t \mathrm {~s}\) and verify that it is stationary when \(t = 3\).
2. Find the position of the particle when \(t = 2\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b9e41fac-9f4b-4165-af03-67ebdcb326de-3_349_987_375_623} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} Particles P and Q move in the same straight line. Particle P starts from rest and has a constant acceleration towards \(Q\) of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Particle \(Q\) starts 125 m from \(P\) at the same time and has a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) away from \(P\). The initial values are shown in Fig. 4.

1. Write down expressions for the distances travelled by P and by Q at time \(t\) seconds after the start of the motion.
2. How much time does it take for P to catch up with Q and how far does P travel in this time?

2 The acceleration of a particle of mass 4 kg is given by \(\mathbf { a } = ( 9 \mathbf { i } - 4 t \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { 2 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors and \(t\) is the time in seconds.

1. Find the acceleration of the particle when \(t = 0\) and also when \(t = 3\).
2. Calculate the force acting on the particle when \(t = 3\). The particle has velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } { } ^ { 1 }\) when \(t = 1\).
3. Find an expression for the velocity of the particle at time \(t\).

3 The position vector, \(r\), of a particle of mass 4 kg at time \(t\) is given by $$\mathbf { r } = t ^ { 2 } \mathbf { i } + \left( 5 t - 2 t ^ { 2 } \right) \mathbf { j }$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors, lengths are in metres and time is in seconds.

1. Find an expression for the acceleration of the particle. The particle is subject to a force \(\mathbf { F }\) and a force \(12 \mathbf { j } \mathbf { N }\).
2. Find \(\mathbf { F }\).

3 A point P on a piece of machinery is moving in a vertical straight line. The displacement of P above ground level at time \(t\) seconds is \(y\) metres. The displacement-time graph for the motion during the time interval \(0 \leqslant t \leqslant 4\) is shown in Fig. 7. \begin{figure}[h] \end{figure}

1. Using the graph, determine for the time interval \(0 \leqslant t \leqslant 4\) (A) the greatest displacement of P above its position when \(t = 0\),
   (B) the greatest distance of P from its position when \(t = 0\),
   (C) the time interval in which P is moving downwards,
   (D) the times when P is instantaneously at rest. The displacement of P in the time interval \(0 \leqslant t \leqslant 3\) is given by \(y = - 4 t ^ { 2 } + 8 t + 12\).
2. Use calculus to find expressions in terms of \(t\) for the velocity and for the acceleration of P in the interval \(0 \leqslant t \leqslant 3\).
3. At what times does P have a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the interval \(0 \leqslant t \leqslant 3\) ? In the time interval \(3 \leqslant t \leqslant 4 , \mathrm { P }\) has a constant acceleration of \(32 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is no sudden change in velocity when \(t = 3\).
4. Find an expression in terms of \(t\) for the displacement of P in the interval \(3 \leqslant t \leqslant 4\).

5 Fig. 3 is a sketch of the velocity-time graph modelling the velocity of a sprinter at the start of a race. \begin{figure}[h] \end{figure}

1. How can you tell from the sketch that the acceleration is not modelled as being constant for \(0 \leqslant t \leqslant 4\) ? The velocity of the sprinter, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), for the time interval \(0 \leqslant t \leqslant 4\) is modelled by the expression $$v = 3 t - \frac { 3 } { 8 } t ^ { 2 } .$$
2. Find the acceleration that the model predicts for \(t = 4\) and comment on what this suggests about the running of the sprinter.
3. Calculate the distance run by the sprinter from \(t = 1\) to \(t = 4\).

5 A particle moves such that at time \(t\) seconds its acceleration is given by $$( 2 \cos t \mathbf { i } - 5 \sin t \mathbf { j } ) \mathrm { m } \mathrm {~s} ^ { - 2 }$$

1. The mass of the particle is 6 kg . Find the magnitude of the resultant force on the particle when \(t = 0\).
2. When \(t = 0\), the velocity of the particle is \(( 2 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find an expression for the velocity of the particle at time \(t\).

2 A particle moves in a straight line and at time \(t\) it has velocity \(v\), where $$v = 3 t ^ { 2 } - 2 \sin 3 t + 6$$

1. 1. Find an expression for the acceleration of the particle at time \(t\).
   2. When \(t = \frac { \pi } { 3 }\), show that the acceleration of the particle is \(2 \pi + 6\).
2. When \(t = 0\), the particle is at the origin. Find an expression for the displacement of the particle from the origin at time \(t\).

4 A particle moves in a horizontal plane under the action of a single force, \(\mathbf { F }\) newtons. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, the position vector, \(\mathbf { r }\) metres, of the particle is given by $$\mathbf { r } = \left( t ^ { 3 } - 3 t ^ { 2 } + 4 \right) \mathbf { i } + \left( 4 t + t ^ { 2 } \right) \mathbf { j }$$

1. Find an expression for the velocity of the particle at time \(t\).
2. The mass of the particle is 3 kg .
   1. Find an expression for \(\mathbf { F }\) at time \(t\).
   2. Find the magnitude of \(\mathbf { F }\) when \(t = 3\).
3. Find the value of \(t\) when \(\mathbf { F }\) acts due north.

1 The velocity of a particle at time \(t\) seconds is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$\mathbf { v } = \left( 4 + 3 t ^ { 2 } \right) \mathbf { i } + ( 12 - 8 t ) \mathbf { j }$$

1. When \(t = 0\), the particle is at the point with position vector \(( 5 \mathbf { i } - 7 \mathbf { j } ) \mathrm { m }\). Find the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).
2. Find the acceleration of the particle at time \(t\).
3. The particle has mass 2 kg . Find the magnitude of the force acting on the particle when \(t = 1\).