3.02f Non-uniform acceleration: using differentiation and integration

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 \(t = 1\).

7 A particle \(P\) is projected from a fixed point \(O\) on a straight line. The displacement \(x\) m of \(P\) from \(O\) at time \(t \mathrm {~s}\) after projection is given by \(x = 0.1 t ^ { 3 } - 0.3 t ^ { 2 } + 0.2 t\).

1. Express the velocity and acceleration of \(P\) in terms of \(t\).
2. Show that when the acceleration of \(P\) is zero, \(P\) is at \(O\).
3. Find the values of \(t\) when \(P\) is stationary. At the instant when \(P\) first leaves \(O\), a particle \(Q\) is projected from \(O\). \(Q\) moves on the same straight line as \(P\) and at time \(t \mathrm {~s}\) after projection the velocity of \(Q\) is given by \(\left( 0.2 t ^ { 2 } - 0.4 \right) \mathrm { ms } ^ { - 1 } . P\) and \(Q\) collide first when \(t = T\).
4. Show that \(T\) satisfies the equation \(t ^ { 2 } - 9 t + 18 = 0\), and hence find \(T\).

7 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-4_122_255_1503_561} The diagram shows two particles \(P\) and \(Q\), of masses 0.2 kg and 0.3 kg respectively, which move on a horizontal surface in the same direction along a straight line. A stationary particle \(R\) of mass 1.5 kg also lies on this line. \(P\) and \(Q\) collide and coalesce to form a combined particle \(C\). Immediately before this collision \(P\) has velocity \(4 \mathrm {~ms} ^ { - 1 }\) and \(Q\) has velocity \(2.5 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the velocity of \(C\) immediately after this collision. At time \(t \mathrm {~s}\) after this collision the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(C\) is given by \(v = V _ { 0 } - 3 t ^ { 2 }\) for \(0 < t \leqslant 0.3\). \(C\) strikes \(R\) when \(t = 0.3\).
2. (a) State the value of \(V _ { 0 }\).
   (b) Calculate the distance \(C\) moves before it strikes \(R\).
   (c) Find the acceleration of \(C\) immediately before it strikes \(R\). Immediately after \(C\) strikes \(R\), the particles have equal speeds but move in opposite directions.
3. Find the speed of \(C\) immediately after it strikes \(R\).

6 A particle \(P\) moves in a straight line. At time \(t\) s after passing through a point \(O\) of the line, the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\). Given that \(x = 0.06 t ^ { 3 } - 0.45 t ^ { 2 } - 0.24 t\), find

1. the velocity and the acceleration of \(P\) when \(t = 0\),
2. the value of \(x\) when \(P\) has its minimum velocity, and the speed of \(P\) at this instant,
3. the positive value of \(t\) when the direction of motion of \(P\) changes.

6 A particle \(P\) moves in a straight line on a horizontal surface. \(P\) passes through a fixed point \(O\) on the line with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after passing through \(O\), the acceleration of \(P\) is \(( 4 + 12 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\).

1. Calculate the velocity of \(P\) when \(t = 3\).
2. Find the distance \(O P\) when \(t = 3\). A second particle \(Q\), having the same mass as \(P\), moves along the same straight line. The displacement of \(Q\) from \(O\) is \(\left( k - 2 t ^ { 3 } \right) \mathrm { m }\), where \(k\) is a constant. When \(t = 3\) the particles collide and coalesce.
3. Find the value of \(k\).
4. Find the common velocity of the particles immediately after their collision.

8 A toy boat moves in a horizontal plane with position vector \(\mathbf { r } = x \mathbf { i } + y \mathbf { j }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors east and north respectively. The origin of the position vectors is at O . The displacements \(x\) and \(y\) are in metres. First consider only the motion of the boat parallel to the \(x\)-axis. For this motion $$x = 8 t - 2 t ^ { 2 }$$ The velocity of the boat in the \(x\)-direction is \(v _ { x } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find an expression in terms of \(t\) for \(v _ { x }\) and determine when the boat instantaneously has zero speed in the \(x\)-direction. Now consider only the motion of the boat parallel to the \(y\)-axis. For this motion $$v _ { y } = ( t - 2 ) ( 3 t - 2 )$$ where \(v _ { y } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the boat in the \(y\)-direction at time \(t\) seconds.
2. Given that \(y = 3\) when \(t = 1\), use integration to show that \(y = t ^ { 3 } - 4 t ^ { 2 } + 4 t + 2\). The position vector of the boat is given in terms of \(t\) by \(\mathbf { r } = \left( 8 t - 2 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 3 } - 4 t ^ { 2 } + 4 t + 2 \right) \mathbf { j }\).
3. Find the time(s) when the boat is due north of O and also the distance of the boat from O at any such times.
4. Find the time(s) when the boat is instantaneously at rest. Find the distance of the boat from O at any such times.
5. Plot a graph of the path of the boat for \(0 \leqslant t \leqslant 2\).

2 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 \(t = 2\).

6 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 \(t \mathrm {~s}\). Show that when \(t = T _ { 1 } , x = 20\) and find the value of \(x\) when \(t = T _ { 2 }\). Section B (36 marks) \(7 \quad\) Abi and Bob are standing on the ground and are trying to raise a small object of weight 250 N to the top of a building. They are using long light ropes. Fig. 7.1 shows the initial situation. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-4_773_1071_429_497} \captionsetup{labelformat=empty} \caption{Fig. 7.1}
   \end{figure} Abi pulls vertically downwards on the rope A with a force \(F\) N. This rope passes over a small smooth pulley and is then connected to the object. Bob pulls on another rope, B, in order to keep the object away from the side of the building. In this situation, the object is stationary and in equilibrium. The tension in rope B, which is horizontal, is 25 N . The pulley is 30 m above the object. The part of rope A between the pulley and the object makes an angle \(\theta\) with the vertical.

8 Fig. 8.1 shows a sledge of mass 40 kg . It is being pulled across a horizontal surface of deep snow by a light horizontal rope. There is a constant resistance to its motion. The tension in the rope is 120 N . \begin{figure}[h] \end{figure} The sledge is initially at rest. After 10 seconds its speed is \(5 \mathrm {~ms} ^ { - 1 }\).

1. Show that the resistance to motion is 100 N . When the speed of the sledge is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the rope breaks. The resistance to motion remains 100 N .
2. Find the speed of the sledge
   (A) 1.6 seconds after the rope breaks,
   (B) 6 seconds after the rope breaks. The sledge is then pushed to the bottom of a ski slope. This is a plane at an angle of \(15 ^ { \circ }\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{83e69140-4abf-4713-85da-922ce7530e47-6_259_853_1457_607} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure} The sledge is attached by a light rope to a winch at the top of the slope. The rope is parallel to the slope and has a constant tension of 200 N . Fig. 8.2 shows the situation when the sledge is part of the way up the slope. The ski slope is smooth.
3. Show that when the sledge has moved from being at rest at the bottom of the slope to the point when its speed is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it has travelled a distance of 13.0 m (to 3 significant figures). When the speed of the sledge is \(8 \mathrm {~ms} ^ { - 1 }\), this rope also breaks.
4. Find the time between the rope breaking and the sledge reaching the bottom of the slope.

1. A particle \(P\) moves along a straight line. The speed of \(P\) at time \(t\) seconds ( \(t \geqslant 0\) ) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \left( p t ^ { 2 } + q t + r \right)\) and \(p , q\) and \(r\) are constants. When \(t = 2\) the speed of \(P\) has its minimum value. When \(t = 0 , v = 11\) and when \(t = 2 , v = 3\)

Find

1. the acceleration of \(P\) when \(t = 3\)
2. the distance travelled by \(P\) in the third second of the motion.
   

3 A particle \(P\) travels in a straight line. The velocity of \(P\) at time \(t\) seconds after it passes through a fixed point \(A\) is given by \(\left( 0.6 t ^ { 2 } + 3 \right) \mathrm { ms } ^ { - 1 }\). Find

1. the velocity of \(P\) when it passes through \(A\),
2. the displacement of \(P\) from \(A\) when \(t = 1.5\),
3. the velocity of \(P\) when it has acceleration \(6 \mathrm {~ms} ^ { - 2 }\).

5 \includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-3_652_1675_959_187} A particle \(P\) can move in a straight line on a horizontal surface. At time \(t\) seconds the displacement of \(P\) from a fixed point \(A\) on the line is \(x \mathrm {~m}\). The diagram shows the \(( t , x )\) graph for \(P\). In the interval \(0 \leqslant t \leqslant 10\), either the speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\), or \(P\) is at rest.

1. Show by calculation that \(T = 1.75\).
2. State the velocity of \(P\) when
   1. \(t = 2\),
   2. \(t = 8\),
   3. \(t = 9\).
   4. Calculate the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 10\). For \(t > 10\), the displacement of \(P\) from \(A\) is given by \(x = 20 t - t ^ { 2 } - 96\).
   5. Calculate the value of \(t\), where \(t > 10\), for which the speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\).

4 A particle travels in a straight line. The velocity of the particle at time \(t \mathrm {~s}\) after leaving a point \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = k t ^ { 2 } - 4 t + 3$$ The distance travelled by the particle in the first 2 s of its motion is 6 m . You may assume that \(v > 0\) in the first 2s of its motion.

1. Find the value of \(k\).
2. Find the value of the minimum velocity of the particle. You do not need to show that this velocity is a minimum.

11 The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of a car at time \(t \mathrm {~s}\), during the first 20 s of its journey, is given by \(v = k t + 0.03 t ^ { 2 }\), where \(k\) is a constant. When \(t = 20\) the acceleration of the car is \(1.3 \mathrm {~ms} ^ { - 2 }\). For \(t > 20\) the car continues its journey with constant acceleration \(1.3 \mathrm {~ms} ^ { - 2 }\) until its speed reaches \(25 \mathrm {~ms} ^ { - 1 }\).

1. Find the value of \(k\).
2. Find the total distance the car has travelled when its speed reaches \(25 \mathrm {~ms} ^ { - 1 }\).

1. A particle, \(P\), moves along a straight line such that at time \(t\) seconds, \(t \geqslant 0\), the velocity of \(P\), \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is modelled as

$$v = 12 + 4 t - t ^ { 2 }$$ Find

1. the magnitude of the acceleration of \(P\) when \(P\) is at instantaneous rest,
2. the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 3\)

1. A particle \(P\) moves along a straight line such that at time \(t\) seconds, \(t \geqslant 0\), after leaving the point \(O\) on the line, the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is modelled as

$$v = ( 7 - 2 t ) ( t + 2 )$$

1. Find the value of \(t\) at the instant when \(P\) stops accelerating.
2. Find the distance of \(P\) from \(O\) at the instant when \(P\) changes its direction of motion. In this question, solutions relying on calculator technology are not acceptable.

1. A fixed point \(O\) lies on a straight line.

A particle \(P\) moves along the straight line.
At time \(t\) seconds, \(t \geqslant 0\), the distance, \(s\) metres, of \(P\) from \(O\) is given by $$s = \frac { 1 } { 3 } t ^ { 3 } - \frac { 5 } { 2 } t ^ { 2 } + 6 t$$

1. Find the acceleration of \(P\) at each of the times when \(P\) is at instantaneous rest.
2. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\)

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

A fixed point \(O\) lies on a straight line.
A particle \(P\) moves along the straight line such that at time \(t\) seconds, \(t \geqslant 0\), after passing through \(O\), the velocity of \(P , v \mathrm {~ms} ^ { - 1 }\), is modelled as $$v = 15 - t ^ { 2 } - 2 t$$

1. Verify that \(P\) comes to instantaneous rest when \(t = 3\)
2. Find the magnitude of the acceleration of \(P\) when \(t = 3\)
3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\)

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

A particle is moving along a straight line.
At time t seconds, \(\mathrm { t } > 0\), the velocity of the particle is \(\mathrm { Vms } ^ { - 1 }\), where $$v = 2 t - 7 \sqrt { t } + 6$$

1. Find the acceleration of the particle when \(t = 4\) When \(\mathrm { t } = 1\) the particle is at the point X .
   When \(\mathrm { t } = 2\) the particle is at the point Y . Given that the particle does not come to instantaneous rest in the interval \(1 < \mathrm { t } < 2\)
2. show that \(X Y = \frac { 1 } { 3 } ( 41 - 28 \sqrt { 2 } )\) metres.

1. A particle \(P\) moves along a straight line.

At time \(t\) seconds, the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is modelled as $$v = 10 t - t ^ { 2 } - k \quad t \geqslant 0$$ where \(k\) is a constant.

1. Find the acceleration of \(P\) at time \(t\) seconds. The particle \(P\) is instantaneously at rest when \(t = 6\)
2. Find the other value of \(t\) when \(P\) is instantaneously at rest.
3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 6\)

1. \hspace{0pt} [In this question position vectors are given relative to a fixed origin \(O\) ]

At time \(t\) seconds, where \(t \geqslant 0\), a particle, \(P\), moves so that its velocity \(\mathbf { v ~ m ~ s } ^ { - 1 }\) is given by $$\mathbf { v } = 6 t \mathbf { i } - 5 t ^ { \frac { 3 } { 2 } } \mathbf { j }$$ When \(t = 0\), the position vector of \(P\) is \(( - 20 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m }\).

1. Find the acceleration of \(P\) when \(t = 4\)
2. Find the position vector of \(P\) when \(t = 4\)

1. \hspace{0pt} [In this question, position vectors are given relative to a fixed origin.] At time \(t\) seconds, where \(t > 0\), a particle \(P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) where

$$\mathbf { v } = 3 t ^ { 2 } \mathbf { i } - 6 t ^ { \frac { 1 } { 2 } } \mathbf { j }$$

1. Find the speed of \(P\) at time \(t = 2\) seconds.
2. Find an expression, in terms of \(t , \mathbf { i }\) and \(\mathbf { j }\), for the acceleration of \(P\) at time \(t\) seconds, where \(t > 0\) At time \(t = 4\) seconds, the position vector of \(P\) is ( \(\mathbf { i } - 4 \mathbf { j }\) ) m.
3. Find the position vector of \(P\) at time \(t = 1\) second.

1. 1. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) moves so that its acceleration a \(\mathrm { ms } ^ { - 2 }\) is given by

$$\mathbf { a } = ( 1 - 4 t ) \mathbf { i } + \left( 3 - t ^ { 2 } \right) \mathbf { j }$$ At the instant when \(t = 0\), the velocity of \(P\) is \(36 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. Find the velocity of \(P\) when \(t = 4\)
2. Find the value of \(t\) at the instant when \(P\) is moving in a direction perpendicular to i
   (ii) At time \(t\) seconds, where \(t \geqslant 0\), a particle \(Q\) moves so that its position vector \(\mathbf { r }\) metres, relative to a fixed origin \(O\), is given by $$\mathbf { r } = \left( t ^ { 2 } - t \right) \mathbf { i } + 3 t \mathbf { j }$$ Find the value of \(t\) at the instant when the speed of \(Q\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. At time \(t\) seconds, a particle \(P\) has velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where

$$\mathbf { v } = 3 t ^ { \frac { 1 } { 2 } } \mathbf { i } - 2 t \mathbf { j } \quad t > 0$$

1. Find the acceleration of \(P\) at time \(t\) seconds, where \(t > 0\)
2. Find the value of \(t\) at the instant when \(P\) is moving in the direction of \(\mathbf { i } - \mathbf { j }\) At time \(t\) seconds, where \(t > 0\), the position vector of \(P\), relative to a fixed origin \(O\), is \(\mathbf { r }\) metres. When \(t = 1 , \mathbf { r } = - \mathbf { j }\)
3. Find an expression for \(\mathbf { r }\) in terms of \(t\).
4. Find the exact distance of \(P\) from \(O\) at the instant when \(P\) is moving with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

10 \includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-6_670_1106_797_258} The diagram shows the velocity-time graph modelling the velocity of a car as it approaches, and drives through, a residential area. The velocity of the car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds for the time interval \(0 \leqslant t \leqslant 5\) is modelled by the equation \(v = p t ^ { 2 } + q t + r\), where \(p , q\) and \(r\) are constants. It is given that the acceleration of the car is zero at \(t = 5\) and the speed of the car then remains constant.

1. Determine the values of \(p , q\) and \(r\).
2. Calculate the distance travelled by the car from \(t = 2\) to \(t = 10\).