3.02a Kinematics language: position, displacement, velocity, acceleration

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

$$v = ( t - 2 ) ( 3 t - 10 ) , \quad t \geqslant 0$$ When \(t = 0 , P\) is at the origin \(O\).

1. Find the acceleration of \(P\) at time \(t\) seconds.
2. Find the total distance travelled by \(P\) in the first 2 seconds of its motion.
3. Show that \(P\) never returns to \(O\), explaining your reasoning.

1. A car of mass 600 kg is moving along a straight horizontal road.

At the instant when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car is modelled as a force of magnitude \(( 200 + 2 v ) \mathrm { N }\). The engine of the car is working at a constant rate of 12 kW .

1. Find the acceleration of the car at the instant when \(v = 20\) Later on the car is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\) At the instant when the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude ( \(200 + 2 v ) \mathrm { N }\). The engine is again working at a constant rate of 12 kW .
   At the instant when the car has speed \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the car is decelerating at \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the value of \(w\).

8 At time \(t\) seconds a particle \(P\) has position vector \(\mathbf { r }\) metres, with respect to a fixed origin \(O\), where $$\mathbf { r } = \left( 4 t ^ { 2 } - k t + 5 \right) \mathbf { i } + \left( 4 t ^ { 3 } + 2 k t ^ { 2 } - 8 t \right) \mathbf { j } , \quad t \geqslant 0 .$$ When \(t = 2 , P\) is moving parallel to the vector \(\mathbf { i }\).

1. Show that \(k = - 5\).
2. Find the values of \(t\) when the magnitude of the acceleration of \(P\) is \(10 \mathrm {~ms} ^ { - 2 }\).

4 Water flows in a constant direction at a constant speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A boat travels in the water at a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the water.

1. The direction in which the boat travels relative to the water is perpendicular to the direction of motion of the water. The resultant velocity of the boat is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(74 ^ { \circ }\) to the direction of motion of the water, as shown in the diagram. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_120_164_662_488} \captionsetup{labelformat=empty} \caption{Velocity of the water}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_126_186_667_890} \captionsetup{labelformat=empty} \caption{Velocity of the boat relative to the water}
   \end{figure}
   1. Find \(V\).
   2. Show that \(u = 3.44\), correct to three significant figures.
2. The boat changes course so that it travels relative to the water at an angle of \(45 ^ { \circ }\) to the direction of motion of the water. The resultant velocity of the boat is now of magnitude \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of the water is unchanged, as shown in the diagram below. $$\xrightarrow { 3.44 \mathrm {~m} \mathrm {~s} ^ { - 1 } }$$
   \includegraphics[max width=\textwidth, alt={}]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_132_273_1493_895}
   Velocity of the boat relative to the water \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_232_355_1498_1384} Find the value of \(v\).
   (4 marks)

6 A van moves from rest on a straight horizontal road.

1. In a simple model, the first 30 seconds of the motion are represented by three separate stages, each lasting 10 seconds and each with a constant acceleration. During the first stage, the van accelerates from rest to a velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   During the second stage, the van accelerates from \(4 \mathrm {~ms} ^ { - 1 }\) to \(12 \mathrm {~ms} ^ { - 1 }\).
   During the third stage, the van accelerates from \(12 \mathrm {~ms} ^ { - 1 }\) to \(16 \mathrm {~ms} ^ { - 1 }\).
   1. Sketch a velocity-time graph to represent the motion of the van during the first 30 seconds of its motion.
   2. Find the total distance that the van travels during the 30 seconds.
   3. Find the average speed of the van during the 30 seconds.
   4. Find the greatest acceleration of the van during the 30 seconds.
2. In another model of the 30 seconds of the motion, the acceleration of the van is assumed to vary during the first and third stages of the motion, but to be constant during the second stage, as shown in the velocity-time graph below. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-006_554_1138_1432_539} The velocity of the van takes the same values at the beginning and the end of each stage of the motion as in part (a).
   1. State, with a reason, whether the distance travelled by the van during the first 10 seconds of the motion in this model is greater or less than the distance travelled during the same time interval in the model in part (a).
   2. Give one reason why this model represents the motion of the van more realistically than the model in part (a).

4 Water flows in a constant direction at a constant speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A boat travels in the water at a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the water.

1. The direction in which the boat travels relative to the water is perpendicular to the direction of motion of the water. The resultant velocity of the boat is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(74 ^ { \circ }\) to the direction of motion of the water, as shown in the diagram. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c220e6c4-2676-4022-8301-7d720dc082b2-3_120_164_662_488} \captionsetup{labelformat=empty} \caption{Velocity of the water}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c220e6c4-2676-4022-8301-7d720dc082b2-3_126_186_667_890} \captionsetup{labelformat=empty} \caption{Velocity of the boat relative to the water}
   \end{figure}
   1. Find \(V\).
   2. Show that \(u = 3.44\), correct to three significant figures.
2. The boat changes course so that it travels relative to the water at an angle of \(45 ^ { \circ }\) to the direction of motion of the water. The resultant velocity of the boat is now of magnitude \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of the water is unchanged, as shown in the diagram below. $$\xrightarrow { 3.44 \mathrm {~m} \mathrm {~s} ^ { - 1 } }$$
   \includegraphics[max width=\textwidth, alt={}]{c220e6c4-2676-4022-8301-7d720dc082b2-3_132_273_1493_895}
   Velocity of the boat relative to the water \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-3_232_355_1498_1384} Find the value of \(v\).
   (4 marks)

6 A van moves from rest on a straight horizontal road.

1. In a simple model, the first 30 seconds of the motion are represented by three separate stages, each lasting 10 seconds and each with a constant acceleration. During the first stage, the van accelerates from rest to a velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   During the second stage, the van accelerates from \(4 \mathrm {~ms} ^ { - 1 }\) to \(12 \mathrm {~ms} ^ { - 1 }\).
   During the third stage, the van accelerates from \(12 \mathrm {~ms} ^ { - 1 }\) to \(16 \mathrm {~ms} ^ { - 1 }\).
   1. Sketch a velocity-time graph to represent the motion of the van during the first 30 seconds of its motion.
   2. Find the total distance that the van travels during the 30 seconds.
   3. Find the average speed of the van during the 30 seconds.
   4. Find the greatest acceleration of the van during the 30 seconds.
2. In another model of the 30 seconds of the motion, the acceleration of the van is assumed to vary during the first and third stages of the motion, but to be constant during the second stage, as shown in the velocity-time graph below. \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-5_554_1138_1432_539} The velocity of the van takes the same values at the beginning and the end of each stage of the motion as in part (a).
   1. State, with a reason, whether the distance travelled by the van during the first 10 seconds of the motion in this model is greater or less than the distance travelled during the same time interval in the model in part (a).
   2. Give one reason why this model represents the motion of the van more realistically than the model in part (a).

3 A particle moves on a horizontal plane, in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, the position vector of the particle is \(\mathbf { r }\) metres, where $$\mathbf { r } = \left( 2 \mathrm { e } ^ { \frac { 1 } { 2 } t } - 8 t + 5 \right) \mathbf { i } + \left( t ^ { 2 } - 6 t \right) \mathbf { j }$$

1. Find an expression for the velocity of the particle at time \(t\).
2. 1. Find the speed of the particle when \(t = 3\).
   2. State the direction in which the particle is travelling when \(t = 3\).
3. Find the acceleration of the particle when \(t = 3\).
4. The mass of the particle is 7 kg . Find the magnitude of the resultant force on the particle when \(t = 3\).

5 A particle moves on a horizontal plane in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, the particle's position vector, \(\mathbf { r }\) metres, is given by $$\mathbf { r } = 8 \left( \cos \frac { 1 } { 4 } t \right) \mathbf { i } - 8 \left( \sin \frac { 1 } { 4 } t \right) \mathbf { j }$$

1. Find an expression for the velocity of the particle at time \(t\).
2. Show that the speed of the particle is a constant.
3. Prove that the particle is moving in a circle.
4. Find the angular speed of the particle.
5. Find an expression for the acceleration of the particle at time \(t\).
6. State the magnitude of the acceleration of the particle.

1 A particle moves under the action of a force, \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), of the particle is given by $$\mathbf { v } = \left( t ^ { 3 } - 15 t - 5 \right) \mathbf { i } + \left( 6 t - t ^ { 2 } \right) \mathbf { j }$$

1. Find an expression for the acceleration of the particle at time \(t\).
2. The mass of the particle is 4 kg .
   1. Show that, at time \(t\), $$\mathbf { F } = \left( 12 t ^ { 2 } - 60 \right) \mathbf { i } + ( 24 - 8 t ) \mathbf { j }$$
   2. Find the magnitude of \(\mathbf { F }\) when \(t = 2\).

3 A fishing boat is travelling between two ports, \(A\) and \(B\), on the shore of a lake. The bearing of \(B\) from \(A\) is \(130 ^ { \circ }\). The fishing boat leaves \(A\) and travels directly towards \(B\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A patrol boat on the lake is travelling with speed \(4 \mathrm {~ms} ^ { - 1 }\) on a bearing of \(040 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-3_713_1319_443_406}

1. Find the velocity of the fishing boat relative to the patrol boat, giving your answer as a speed together with a bearing.
2. When the patrol boat is 1500 m due west of the fishing boat, it changes direction in order to intercept the fishing boat in the shortest possible time.
   1. Find the bearing on which the patrol boat should travel in order to intercept the fishing boat.
   2. Given that the patrol boat intercepts the fishing boat before it reaches \(B\), find the time, in seconds, that it takes the patrol boat to intercept the fishing boat after changing direction.
   3. State a modelling assumption necessary for answering this question, other than the boats being particles.

18 In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors representing due east and due north respectively. A particle, \(T\), is moving on a plane at a constant speed.
The path followed by \(T\) makes the exact shape of a triangle \(A B C\). \(T\) moves around \(A B C\) in an anticlockwise direction as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-28_447_366_671_925} On its journey from \(A\) to \(B\) the velocity vector of \(T\) is \(( 3 \mathbf { i } + \sqrt { 3 } \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) 18

1. Find the speed of \(T\) as it moves from \(A\) to \(B\) 18
2. On its journey from \(B\) to \(C\) the velocity vector of \(T\) is \(( - 3 \mathbf { i } + \sqrt { 3 } \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) Show that the acute angle \(A B C = 60 ^ { \circ }\) 18
3. It is given that \(A B C\) is an equilateral triangle. \(T\) returns to its initial position after 9 seconds.
   Vertex \(B\) lies at position vector \(\left[ \begin{array} { l } 1 \\ 0 \end{array} \right]\) metres with respect to a fixed origin \(O\) Find the position vector of \(C\)

6 A particle travels along a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after \(t\) seconds is given by $$v = t ^ { 3 } - 6 t ^ { 2 } + 8 t \text { for } 0 \leqslant t \leqslant 4$$ When \(t = 0\) the particle is at rest at the point \(P\).

1. Find the times (other than \(t = 0\) ) when the particle is at rest. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 4\).
2. Find the acceleration of the particle when \(t = 2\).
3. Find an expression for the displacement of the particle from \(P\) after \(t\) seconds. Hence state its displacement from \(P\) when \(t = 2\) and find its average speed between \(t = 0\) and \(t = 2\).

7 At a given instant two stunt cars, \(X\) and \(Y\), are at distances 500 m and 800 m respectively from the point of intersection, \(O\), of two straight roads crossing at right angles. The stunt cars are approaching \(O\) at uniform speeds of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, one on each road. Find, in either order,

1. the time taken to reach the point of closest approach,
2. the shortest distance between the stunt cars.

11 In a training exercise, a submarine is travelling due north at \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). The submarine commander sees his target 5 km away on a bearing of \(310 ^ { \circ }\). The target is travelling due east at \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).

1. If each of the submarine and target maintains its present course and speed, find the shortest distance between them.
2. In fact, as soon as he sees the target, the submarine commander changes course, without changing speed, so as to intercept the target as quickly as possible. Find
   1. the course, in degrees, set by the submarine commander,
   2. the time taken, in minutes, to intercept the target from the moment that the course changes.

7 A particle travels along a straight line. Its velocity \(v \mathrm {~ms} ^ { - 1 }\) after \(t\) seconds is given by $$v = t ^ { 3 } - 9 t ^ { 2 } + 20 t$$ When \(t = 0\), the particle is at rest at \(P\).

1. Find the times, other than \(t = 0\), at which the particle is at rest.
2. Find the displacement of the particle from \(P\) when \(t = 2\).

A particle \(P\) moves along the \(x\)-axis in the positive direction. The velocity of \(P\) at time \(t \text{ s}\) is \(0.03t^2 \text{ m s}^{-1}\). When \(t = 5\) the displacement of \(P\) from the origin \(O\) is \(2.5 \text{ m}\).

1. Find an expression, in terms of \(t\), for the displacement of \(P\) from \(O\). [4]
2. Find the velocity of \(P\) when its displacement from \(O\) is \(11.25 \text{ m}\). [3]

A vehicle is moving in a straight line. The velocity \(v \text{ m s}^{-1}\) at time \(t \text{ s}\) after the vehicle starts is given by $$v = A(t - 0.05t^2) \text{ for } 0 \leq t \leq 15,$$ $$v = \frac{B}{t} \text{ for } t \geq 15,$$ where \(A\) and \(B\) are constants. The distance travelled by the vehicle between \(t = 0\) and \(t = 15\) is \(225 \text{ m}\).

1. Find the value of \(A\) and show that \(B = 3375\). [5]
2. Find an expression in terms of \(t\) for the total distance travelled by the vehicle when \(t \geq 15\). [3]
3. Find the speed of the vehicle when it has travelled a total distance of \(315 \text{ m}\). [3]

A particle moves in a straight line. At time \(t\) seconds, its displacement from a fixed point is \(s\) metres, where $$s = t^3 - 6t^2 + 9t$$

1. Find expressions for the velocity and acceleration of the particle at time \(t\). [4]
2. Find the times when the particle is at rest. [2]

A particle \(P\) moves on a straight line. It starts at a point \(O\) on the line and returns to \(O\) 100 s later. The velocity of \(P\) is \(v \text{ m s}^{-1}\) at time \(t\) s after leaving \(O\), where $$v = 0.0001t^3 - 0.015t^2 + 0.5t.$$

1. Show that \(P\) is instantaneously at rest when \(t = 0\), \(t = 50\) and \(t = 100\). [2]
2. Find the values of \(v\) at the times for which the acceleration of \(P\) is zero, and sketch the velocity-time graph for \(P\)'s motion for \(0 \leq t \leq 100\). [7]
3. Find the greatest distance of \(P\) from \(O\) for \(0 \leq t \leq 100\). [4]

A particle \(A\) moves in a straight line with constant speed \(10\) m s\(^{-1}\). Two seconds after \(A\) passes a point \(O\) on the line, a particle \(B\) passes through \(O\), moving along the line in the same direction as \(A\). Particle \(B\) has speed \(16\) m s\(^{-1}\) at \(O\) and has a constant deceleration of \(2\) m s\(^{-2}\).

1. Find expressions, in terms of \(t\), for the displacement from \(O\) of each particle \(t\) s after \(B\) passes through \(O\). [3]
2. Find the distance between the particles when \(B\) comes to instantaneous rest. [3]
3. Find the minimum distance between the particles. [3]

A particle \(P\) moves in a straight line starting from a point \(O\). At time \(t \text{ s}\) after leaving \(O\), the displacement \(s \text{ m}\) from \(O\) is given by \(s = t^3 - 4t^2 + 4t\) and the velocity is \(v \text{ m s}^{-1}\).

1. Find an expression for \(v\) in terms of \(t\). [2]
2. Find the two values of \(t\) for which \(P\) is at instantaneous rest. [2]
3. Find the minimum velocity of \(P\). [3]

A particle \(P\) moves in a straight line starting from a point \(O\). The velocity \(v\text{ m s}^{-1}\) of \(P\) at time \(t\text{ s}\) is given by $$v = 12t - 4t^2 \quad \text{for } 0 \leqslant t \leqslant 2,$$ $$v = 16 - 4t \quad \text{for } 2 \leqslant t \leqslant 4.$$

1. Find the maximum velocity of \(P\) during the first \(2\text{ s}\). [3]
2. Determine, with justification, whether there is any instantaneous change in the acceleration of \(P\) when \(t = 2\). [2]
3. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 4\). [3]
4. Find the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\). [5]

Particles \(P\) and \(Q\) leave a fixed point \(A\) at the same time and travel in the same straight line. The velocity of \(P\) after \(t\) seconds is \(6(t - 3)\) m s\(^{-1}\) and the velocity of \(Q\) after \(t\) seconds is \((10 - 2t)\) m s\(^{-1}\).

1. Sketch, on the same axes, velocity-time graphs for \(P\) and \(Q\) for \(0 \leq t \leq 5\). [3]
2. Verify that \(P\) and \(Q\) meet after 5 seconds. [4]
3. Find the greatest distance between \(P\) and \(Q\) for \(0 \leq t \leq 5\). [4]