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

A van of mass 1500 kg is driving up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{12}\). The resistance to motion due to non-gravitational forces is modelled as a constant force of magnitude 1000 N. Given that initially the speed of the van is 30 m s\(^{-1}\) and that the van's engine is working at a rate of 60 kW,

1. calculate the magnitude of the initial deceleration of the van. [4]

When travelling up the same hill, the rate of working of the van's engine is increased to 80 kW. Using the same model for the resistance due to non-gravitational forces,

1. calculate in m s\(^{-1}\) the constant speed which can be sustained by the van at this rate of working. [4]
2. Give one reason why the use of this model for resistance may mean that your answer to part (b) is too high. [1]

A particle \(P\) of mass 0.3 kg is moving under the action of a single force \(\mathbf{F}\) newtons. At time \(t\) seconds the velocity of \(P\), \(\mathbf{v}\) m s\(^{-1}\), is given by $$\mathbf{v} = 3t\mathbf{i} + (6t - 4)\mathbf{j}.$$

1. Calculate, to 3 significant figures, the magnitude of \(\mathbf{F}\) when \(t = 2\). [5]

When \(t = 0\), \(P\) is at the point \(A\). The position vector of \(A\) with respect to a fixed origin \(O\) is \((3\mathbf{i} - 4\mathbf{j})\) m. When \(t = 4\), \(P\) is at the point \(B\).

1. Find the position vector of \(B\). [5]

A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\) at time \(t\) seconds is \((4t - 8)\) m s\(^{-2}\), measured in the direction of \(x\) increasing. The velocity of \(P\) at time \(t\) seconds is \(v\) m s\(^{-1}\). Given that \(v = 6\) when \(t = 0\), find

1. \(v\) in terms of \(t\), [4]
2. the distance between the two points where \(P\) is instantaneously at rest. [7]

A particle \(P\) of mass 0.4 kg is moving so that its position vector \(\mathbf{r}\) metres at time \(t\) seconds is given by $$\mathbf{r} = (t^2 + 4t)\mathbf{i} + (3t - t^3)\mathbf{j}.$$

1. Calculate the speed of \(P\) when \(t = 3\). [5]

When \(t = 3\), the particle \(P\) is given an impulse \((8\mathbf{i} - 12\mathbf{j})\) N s.

1. Find the velocity of \(P\) immediately after the impulse. [3]

At time \(t\) seconds \((t \geq 0)\), a particle \(P\) has position vector \(\mathbf{p}\) metres, with respect to a fixed origin \(O\), where $$\mathbf{p} = (3t^2 - 6t + 4)\mathbf{i} + (3t^3 - 4t)\mathbf{j}.$$ Find

1. the velocity of \(P\) at time \(t\) seconds, [2]
2. the value of \(t\) when \(P\) is moving parallel to the vector \(\mathbf{i}\). [3]

When \(t = 1\), the particle \(P\) receives an impulse of \((2\mathbf{i} - 6\mathbf{j})\) N s. Given that the mass of \(P\) is 0.5 kg,

1. find the velocity of \(P\) immediately after the impulse. [4]

A particle \(P\) is moving in a plane. At time \(t\) seconds, \(P\) is moving with velocity \(\mathbf{v}\) m s\(^{-1}\), where \(\mathbf{v} = 2t\mathbf{i} - 3t^2\mathbf{j}\). Find

1. the speed of \(P\) when \(t = 4\) [2]
2. the acceleration of \(P\) when \(t = 4\) [3]

Given that \(P\) is at the point with position vector \((-4\mathbf{i} + \mathbf{j})\) m when \(t = 1\),

1. find the position vector of \(P\) when \(t = 4\) [5]

A boy starts at the corner \(O\) of a rectangular playing field and runs across the field with constant velocity vector \((\mathbf{i} + 2\mathbf{j})\) ms\(^{-1}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in the directions of two perpendicular sides of the field. After 40 seconds, at the point \(P\) in the field, he changes speed and direction so that his new velocity vector is \((2.4\mathbf{i} - 1.8\mathbf{j})\) ms\(^{-1}\) and maintains this velocity until he reaches the point \(Q\), where \(PQ = 75\) m. Calculate

1. the distance \(OP\), [3 marks]
2. the time taken to travel from \(P\) to \(Q\), [2 marks]
3. the position vector of \(Q\) relative to \(O\). [3 marks]

Another boy travels directly from \(O\) to \(Q\) with constant velocity \((a\mathbf{i} + b\mathbf{j})\) ms\(^{-1}\), leaving \(O\) and reaching \(Q\) at the same times as the first boy.

1. Find the values of the constants \(a\) and \(b\). [4 marks]

Relative to a fixed origin \(O\), the points \(X\) and \(Y\) have position vectors \((4\mathbf{i} - 5\mathbf{j})\) m and \((12\mathbf{i} + \mathbf{j})\) m respectively, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors.

1. Find the distance \(XY\). [2 marks]

A particle \(P\) of mass \(2\) kg moves from \(X\) to \(Y\) in \(4\) seconds, in a straight line at a constant speed.

1. Show that the velocity vector of \(P\) is \((2\mathbf{i} + 1.5\mathbf{j}) \text{ ms}^{-1}\). [3 marks]

The particle continues beyond \(Y\) with the same constant velocity.

1. Write down an expression for the position vector of \(P\) \(t\) seconds after leaving \(X\). [2 marks]
2. Find the value of \(t\) when \(P\) is at the point with position vector \((16\mathbf{i} + 4\mathbf{j})\) m. [2 marks]

When it is moving with the same constant speed, \(P\) collides directly with another particle \(Q\), of mass \(4\) kg, which is at rest. \(P\) and \(Q\) coalesce and move together as a single particle.

1. Find the velocity vector of the combined particle after the collision. [5 marks]

\includegraphics{figure_7} A sprinter \(S\) starts from rest at time \(t = 0\), where \(t\) is in seconds, and runs in a straight line. For \(0 \leq t \leq 3\), \(S\) has velocity \((6t - t^2)\) m s\(^{-1}\). For \(3 < t \leq 22\), \(S\) runs at a constant speed of \(9\) m s\(^{-1}\). For \(t > 22\), \(S\) decelerates at \(0.6\) m s\(^{-2}\) (see diagram).

1. Express the acceleration of \(S\) during the first \(3\) seconds in terms of \(t\). [2]
2. Show that \(S\) runs \(18\) m in the first \(3\) seconds of motion. [5]
3. Calculate the time \(S\) takes to run \(100\) m. [3]
4. Calculate the time \(S\) takes to run \(200\) m. [7]

Two trains \(A\) and \(B\) leave the same station, \(O\), at 10 a.m. and travel along straight horizontal tracks. \(A\) travels with constant speed \(80 \text{ km h}^{-1}\) due east and \(B\) travels with constant speed \(52 \text{ km h}^{-1}\) in the direction \((5\mathbf{i} + 12\mathbf{j})\) where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors due east and due north respectively.

1. Show that the velocity of \(B\) is \((20\mathbf{i} + 48\mathbf{j}) \text{ km h}^{-1}\). [3 marks]
2. Find the displacement vector of \(B\) from \(A\) at 10:15 a.m. [3 marks] Given that the trains are 23 km apart \(t\) minutes after 10 a.m.
3. find the value of \(t\) correct to the nearest whole number. [6 marks]

The displacement, \(x\) m, from the origin O of a particle on the \(x\)-axis is given by $$x = 10 + 36t + 3t^2 - 2t^3,$$ where \(t\) is the time in seconds and \(-4 \leqslant t \leqslant 6\).

1. Write down the displacement of the particle when \(t = 0\). [1]
2. Find an expression in terms of \(t\) for the velocity, \(v\) ms\(^{-1}\), of the particle. [2]
3. Find an expression in terms of \(t\) for the acceleration of the particle. [2]
4. Find the maximum value of \(v\) in the interval \(-4 \leqslant t \leqslant 6\). [3]
5. Show that \(v = 0\) only when \(t = -2\) and when \(t = 3\). Find the values of \(x\) at these times. [5]
6. Calculate the distance travelled by the particle from \(t = 0\) to \(t = 4\). [3]
7. Determine how many times the particle passes through O in the interval \(-4 \leqslant t \leqslant 6\). [3]

A particle moves along the \(x\)-axis with velocity, \(v\) ms\(^{-1}\), at time \(t\) given by $$v = 24t - 6t^2.$$ The positive direction is in the sense of \(x\) increasing.

1. Find an expression for the acceleration of the particle at time \(t\). [2]
2. Find the times, \(t_1\) and \(t_2\), at which the particle has zero speed. [2]
3. Find the distance travelled between the times \(t_1\) and \(t_2\). [4]

Two girls, Marie and Nina, are members of an Olympic hockey team. They are doing fitness training. Marie runs along a straight line at a constant speed of \(6\) ms\(^{-1}\). Nina is stationary at a point O on the line until Marie passes her. Nina immediately runs after Marie until she catches up with her. The time, \(t\) s, is measured from the moment when Nina starts running. So when \(t = 0\), both girls are at O. Nina's acceleration, \(a\) ms\(^{-2}\), is given by \begin{align} a &= 4 - t \quad \text{for } 0 < t < 4,
a &= 0 \quad \text{for } t > 4. \end{align}

1. Show that Nina's speed, \(v\) ms\(^{-1}\), is given by \begin{align} v &= 4t - \frac{1}{2}t^2 \quad \text{for } 0 < t < 4,
   v &= 8 \quad \text{for } t > 4. \end{align} [3]
2. Find an expression for the distance Nina has run at time \(t\), for \(0 \leqslant t < 4\). Find how far Nina has run when \(t = 4\) and when \(t = 5\frac{1}{4}\). [4]
3. Show that Nina catches up with Marie when \(t = 5\frac{1}{4}\). [1]

The velocity, \(v\) ms\(^{-1}\), of a particle moving along a straight line is given by $$v = 3t^2 - 12t + 14,$$ where \(t\) is the time in seconds.

1. Find an expression for the acceleration of the particle at time \(t\). [2]
2. Find the displacement of the particle from its position when \(t = 1\) to its position when \(t = 3\). [4]
3. You are given that \(v\) is always positive. Explain how this tells you that the distance travelled by the particle between \(t = 1\) and \(t = 3\) has the same value as the displacement between these times. [2]

Fig. 4 shows the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) in the directions of the cartesian axes \(Ox\) and \(Oy\), respectively. O is the origin of the axes and of position vectors. \includegraphics{figure_1} The position vector of a particle is given by \(\mathbf{r} = 3t\mathbf{i} + (18t^2 - 11)\mathbf{j}\) for \(t \geq 0\), where \(t\) is time.

1. Show that the path of the particle cuts the \(x\)-axis just once. [2]
2. Find an expression for the velocity of the particle at time \(t\). Deduce that the particle never travels in the \(\mathbf{j}\) direction. [3]
3. Find the cartesian equation of the path of the particle, simplifying your answer. [3]

At time \(t\) seconds, a particle has position with respect to an origin O given by the vector $$\mathbf{r} = \begin{pmatrix} 8t \\ 10t^2 - 2t^3 \end{pmatrix},$$ where \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) and \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\) are perpendicular unit vectors east and north respectively and distances are in metres.

1. When \(t = 1\), the particle is at P. Find the bearing of P from O. [2]
2. Find the velocity of the particle at time \(t\) and show that it is never zero. [3]
3. Determine the time(s), if any, when the acceleration of the particle is zero. [3]

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 $$x = 8t - 2t^2.$$ The velocity of the boat in the \(x\)-direction is \(v_x\) ms\(^{-1}\).

1. Find an expression in terms of \(t\) for \(v_x\) and determine when the boat instantaneously has zero speed in the \(x\)-direction. [3]

Now consider only the motion of the boat parallel to the \(y\)-axis. For this motion $$v_y = (t - 2)(3t - 2),$$ where \(v_y\) ms\(^{-1}\) is the velocity of the boat in the \(y\)-direction at time \(t\) seconds.

1. Given that \(y = 3\) when \(t = 1\), use integration to show that \(y = t^3 - 4t^2 + 4t + 2\). [4]

The position vector of the boat is given in terms of \(t\) by \(\mathbf{r} = (8t - 2t^2)\mathbf{i} + (t^3 - 4t^2 + 4t + 2)\mathbf{j}\).

1. 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]
2. Find the time(s) when the boat is instantaneously at rest. Find the distance of the boat from O at any such times. [5]
3. Plot a graph of the path of the boat for \(0 \leq t \leq 2\). [3]

A particle \(P\) starts from the point \(O\) and moves such that its position vector \(\mathbf{r}\) m relative to \(O\) after \(t\) seconds is given by \(\mathbf{r} = at^2\mathbf{i} + bt\mathbf{j}\). 60 seconds after \(P\) leaves \(O\) it is at the point \(Q\) with position vector \((90\mathbf{i} + 30\mathbf{j})\) m.

1. Find the values of the constants \(a\) and \(b\). [3 marks]
2. Find the speed of \(P\) when it is at \(Q\). [4 marks]
3. Sketch the path followed by \(P\) for \(0 \leq t \leq 60\). [2 marks]

Relative to a fixed origin \(O\), the points \(X\) and \(Y\) have position vectors \((4\mathbf{i} - 5\mathbf{j})\) m and \((12\mathbf{i} + \mathbf{j})\) m respectively, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in the directions due east and due north respectively. A particle \(P\) starts from \(X\), and \(t\) seconds later its position vector relative to \(O\) is \((2t + 4)\mathbf{i} + (kt^2 - 5)\mathbf{j}\).

1. Find the value of \(k\) if \(P\) takes \(4\) seconds to reach \(Y\). [3 marks]
2. Show that \(P\) has constant acceleration and find the magnitude and direction of this acceleration. [4 marks]

A car of mass 1500 kg travels along a straight horizontal road. The resistance to the motion of the car is \(kv^{\frac{3}{2}}\) N, where \(v\) ms\(^{-1}\) is the speed of the car and \(k\) is a constant. At the instant when the engine produces a power of 15000 W, the car has speed 15 ms\(^{-1}\) and is accelerating at 0.4 ms\(^{-2}\).

1. Find the value of \(k\). [4]

It is given that the greatest steady speed of the car on this road is 30 ms\(^{-1}\).

1. Find the greatest power that the engine can produce. [3]

A river of width 40 m flows with uniform and constant speed between straight banks. A swimmer crosses as quickly as possible and takes 30 s to reach the other side. She is carried 25 m downstream. Find

1. the speed of the river, [2]
2. the speed of the swimmer relative to the water. [2]

A pilot flying an aircraft at a constant speed of 2000 kmh\(^{-1}\) detects an enemy aircraft 100 km away on a bearing of 045°. The enemy aircraft is flying at a constant velocity of 1500 kmh\(^{-1}\) due west. Find

1. the course, as a bearing to the nearest degree, that the pilot should set up in order to intercept the enemy aircraft,
2. the time, to the nearest s, that the pilot will take to reach the enemy aircraft. [11]

A boy enters a large horizontal field and sees a friend 100 m due north. The friend is walking in an easterly direction at a constant speed of 0.75 m s\(^{-1}\). The boy can walk at a maximum speed of 1 m s\(^{-1}\). Find the shortest time for the boy to intercept his friend and the bearing on which he must travel to achieve this. [6]

Boat \(A\) is sailing due east at a constant speed of 10 km h\(^{-1}\). To an observer on \(A\), the wind appears to be blowing from due south. A second boat \(B\) is sailing due north at a constant speed of 14 km h\(^{-1}\). To an observer on \(B\), the wind appears to be blowing from the south west. The velocity of the wind relative to the earth is constant and is the same for both boats. Find the velocity of the wind relative to the earth, stating its magnitude and direction. [7]

\includegraphics{figure_2} Boat \(A\) is travelling with constant speed 7.9 m s\(^{-1}\) on a course with bearing 035°. Boat \(B\) is travelling with constant speed 10.5 m s\(^{-1}\) on a course with bearing 330°. At one instant, the boats are 1500 m apart with \(B\) on a bearing of 125° from \(A\) (see diagram).

1. Find the magnitude and the bearing of the velocity of \(B\) relative to \(A\). [5]
2. Find the shortest distance between \(A\) and \(B\) in the subsequent motion. [2]
3. Find the time taken from the instant when \(A\) and \(B\) are 1500 m apart to the instant when \(A\) and \(B\) are at the point of closest approach. [2]