1.10h Vectors in kinematics: uniform acceleration in vector form

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]

A cyclist \(C\) is moving with a constant speed of \(10\) m s\(^{-1}\) due south. Cyclist \(D\) is moving with a constant speed of \(16\) m s\(^{-1}\) on a bearing of \(240°\).

1. Show that the magnitude of the velocity of \(C\) relative to \(D\) is \(14\) m s\(^{-1}\). [3]

At \(2\) pm, \(D\) is \(4\) km due east of \(C\).

1. Find
   1. the shortest distance between \(C\) and \(D\) during the subsequent motion,
   2. the time, to the nearest minute, at which this shortest distance occurs.
   [7]

A particle \(A\) has constant velocity \((3\mathbf{i} + \mathbf{j})\) m s\(^{-1}\) and a particle \(B\) has constant velocity \((\mathbf{i} - \mathbf{k})\) m s\(^{-1}\). At time \(t = 0\) seconds, the position vectors of the particles \(A\) and \(B\) with respect to a fixed origin \(O\) are \((-6\mathbf{i} + 4\mathbf{j} - 3\mathbf{k})\) m and \((-2\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})\) m respectively.

1. Show that, in the subsequent motion, the minimum distance between \(A\) and \(B\) is \(4\sqrt{2}\) m. [6]
2. Find the position vector of \(A\) at the instant when the distance between \(A\) and \(B\) is a minimum. [2]

At noon two ships \(A\) and \(B\) are 20 km apart with \(A\) on a bearing of 230° from \(B\). Ship \(B\) is moving at 6 km h\(^{-1}\) on a bearing of 015°. The maximum speed of \(A\) is 12 km h\(^{-1}\). Ship \(A\) sets a course to intercept \(B\) as soon as possible.

1. Find the course set by \(A\), giving your answer as a bearing to the nearest degree. [4]
2. Find the time at which \(A\) intercepts \(B\). [4]

\includegraphics{figure_1} A girl swims in still water at 1 m s\(^{-1}\). She swims across a river which is 336 m wide and is flowing at 0.6 m s\(^{-1}\). She sets off from a point \(A\) on one bank and lands at a point \(B\), which is directly opposite \(A\), on the other bank as shown in Fig. 1. Find

1. the direction, relative to the earth, in which she swims, [3]
2. the time that she takes to cross the river. [3]

Two horizontal roads cross at right angles. One is directed from south to north, and the other from east to west. A tractor travels north on the first road at a constant speed of 6 m s\(^{-1}\) and at noon is 200 m south of the junction. A car heads west on the second road at a constant speed of 24 m s\(^{-1}\) and at noon is 960 m east of the junction.

1. Find the magnitude and direction of the velocity of the car relative to the tractor. [6]
2. Find the shortest distance between the car and the tractor. [8]

\includegraphics{figure_6} A ship \(P\) is moving with constant velocity 7 m s\(^{-1}\) in the direction with bearing 110°. A second ship \(Q\) is moving with constant speed 10 m s\(^{-1}\) in a straight line. At one instant \(Q\) is at the point \(X\), and \(P\) is 7400 m from \(Q\) on a bearing of 050° (see diagram). In the subsequent motion, the shortest distance between \(P\) and \(Q\) is 1790 m.

1. Show that one possible direction for the velocity of \(Q\) relative to \(P\) has bearing 036°, to the nearest degree, and find the bearing of the other possible direction of this relative velocity. [3]

Given that the velocity of \(Q\) relative to \(P\) has bearing 036°, find

1. the bearing of the direction in which \(Q\) is moving, [4]
2. the magnitude of the velocity of \(Q\) relative to \(P\), [2]
3. the time taken for \(Q\) to travel from \(X\) to the position where the two ships are closest together, [3]
4. the bearing of \(P\) from \(Q\) when the two ships are closest together. [1]

A particle \(P\) moves in a plane so that its position vector, \(\mathbf{r}\) metres at time \(t\) seconds, satisfies the differential equation $$\frac{d\mathbf{r}}{dt} + \mathbf{r} = t\mathbf{i} + e^{-t}\mathbf{j}$$ When \(t = 0\) the particle is at the point with position vector \((\mathbf{i} + \mathbf{j})\) m. Find \(\mathbf{r}\) in terms of \(t\). [9]

A particle \(P\) moves in the \(x\)-\(y\) plane and has position vector \(\mathbf{r}\) metres relative to a fixed origin \(O\) at time \(t\) s. Given that \(\mathbf{r}\) satisfies the vector differential equation $$\frac{d^2\mathbf{r}}{dt^2} + 9\mathbf{r} = 8\sin t \mathbf{i}$$ and that when \(t = 0\) s, \(P\) is at \(O\) and moving with velocity \((\mathbf{i} + 3\mathbf{j})\) m s\(^{-1}\),

1. find \(\mathbf{r}\) at time \(t\). [11]
2. Hence find when \(P\) next returns to \(O\). [2]

\includegraphics{figure_9} The diagram shows a small block \(B\), of mass \(0.2\) kg, and a particle \(P\), of mass \(0.5\) kg, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane. The block can move on the horizontal surface, which is rough. The particle can move on the inclined plane, which is smooth and which makes an angle of \(\theta\) with the horizontal where \(\tan \theta = \frac{3}{4}\). The system is released from rest. In the first \(0.4\) seconds of the motion \(P\) moves \(0.3\) m down the plane and \(B\) does not reach the pulley.

1. Find the tension in the string during the first \(0.4\) seconds of the motion. [4]
2. Calculate the coefficient of friction between \(B\) and the horizontal surface. [5]

A particle \(P\) moves with constant acceleration \((-4\mathbf{i} + 2\mathbf{j})\) ms\(^{-2}\). At time \(t = 0\) seconds, \(P\) is moving with velocity \((7\mathbf{i} + 6\mathbf{j})\) ms\(^{-1}\).

1. Determine the speed of \(P\) when \(t = 3\). [4]
2. Determine the change in displacement of \(P\) between \(t = 0\) and \(t = 3\). [2]

\includegraphics{figure_11} A particle \(P\) moves freely under gravity in the plane of a fixed horizontal axis \(Ox\), which lies on horizontal ground, and a fixed vertical axis \(Oy\). \(P\) is projected from \(O\) with a velocity whose components along \(Ox\) and \(Oy\) are \(U\) and \(V\), respectively. \(P\) returns to the ground at a point \(C\).

1. Determine, in terms of \(U\), \(V\) and \(g\), the distance \(OC\). [4] \includegraphics{figure_11b} \(P\) passes through two points \(A\) and \(B\), each at a height \(h\) above the ground and a distance \(d\) apart, as shown in the diagram.
2. Write down the horizontal and vertical components of the velocity of \(P\) at \(A\). [2]
3. Hence determine an expression for \(d\) in terms of \(U\), \(V\), \(g\) and \(h\). [3]
4. Given that the direction of motion of \(P\) as it passes through \(A\) is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac{1}{2}\), determine an expression for \(V\) in terms of \(g\), \(d\) and \(h\). [4]

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 \((-8\mathbf{i} - 54\mathbf{j})\) N and a variable horizontal force \((4t\mathbf{i} + 6(2t - 1)^2\mathbf{j})\) N.

1. Determine the value of \(t\) when the forces acting on \(P\) are in equilibrium. [2]

It is given that \(P\) is at rest when \(t = 0\).

1. Determine the speed of \(P\) at the instant when \(P\) is moving due north. [6]
2. Determine the distance between the positions of \(P\) when \(t = 0\) and \(t = 3\). [5]

In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. A particle \(P\) is moving on a smooth horizontal surface under the action of a single force \(\mathbf{F}\) N. At time \(t\) seconds, where \(t \geq 0\), the velocity \(\mathbf{v} \mathrm{m s}^{-1}\) of \(P\), relative to a fixed origin \(O\), is given by $$\mathbf{v} = (1 - 2t)\mathbf{i} + (2t^2 + t - 13)\mathbf{j}.$$

1. Show that \(P\) is never stationary. [2]
2. Find, in terms of \(\mathbf{i}\) and \(\mathbf{j}\), the acceleration of \(P\) at time \(t\). [1]

The mass of \(P\) is 0.5 kg.

1. Determine the magnitude of \(\mathbf{F}\) when \(P\) is moving in the direction of the vector \(-2\mathbf{i} + \mathbf{j}\). Give your answer correct to 3 significant figures. [5]

When \(t = 1\), \(P\) is at the point with position vector \(\frac{1}{6}\mathbf{j}\).

1. Determine the bearing of \(P\) from \(O\) at time \(t = 1.5\). [5]

A particle \(P\) moves with constant acceleration \((3\mathbf{i} - 2\mathbf{j}) \text{ms}^{-2}\). At time \(t = 4\) seconds, \(P\) has velocity \(6\mathbf{i} \text{ms}^{-1}\). Determine the speed of \(P\) at time \(t = 0\) seconds. [4]

Two particles \(P\) and \(Q\) are moving in separate straight lines across a smooth horizontal surface. \(P\) moves with constant velocity \((3\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\) \(Q\) moves from position vector \((5\mathbf{i} - 7\mathbf{j})\) metres to position vector \((14\mathbf{i} + 5\mathbf{j})\) metres during a 3 second period.

1. Show that \(P\) and \(Q\) move along parallel lines. [3 marks]
2. Stevie says Q is also moving with a constant velocity of \((3\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\) Explain why Stevie may be incorrect. [1 mark]
3. A third particle \(R\) is moving with a constant speed of 4 m s\(^{-1}\), in a straight line, across the same surface. \(P\) and \(R\) move along lines that intersect at a fixed point \(X\) It is given that: • \(P\) passes through \(X\) exactly 2 seconds after \(R\) passes through \(X\) • \(P\) and \(R\) are exactly 13 metres apart 3 seconds after \(R\) passes through \(X\) Show that \(P\) and \(R\) move along perpendicular lines. [5 marks]

At time \(t = 0\), a parachutist jumps out of an airplane that is travelling horizontally. The velocity, \(\mathbf{v}\) m s\(^{-1}\), of the parachutist at time \(t\) seconds is given by: $$\mathbf{v} = (40e^{-0.2t})\mathbf{i} + 50(e^{-0.2t} - 1)\mathbf{j}$$ The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively. Assume that the parachutist is at the origin when \(t = 0\) Model the parachutist as a particle.

1. Find an expression for the position vector of the parachutist at time \(t\). [4 marks]
2. The parachutist opens her parachute when she has travelled 100 metres horizontally. Find the vertical displacement of the parachutist from the origin when she opens her parachute. [4 marks]
3. Carefully, explaining the steps that you take, deduce the value of \(g\) used in the formulation of this model. [3 marks]

In this question use \(g = 9.81\) m s\(^{-2}\). A ball is projected from the origin. After 2.5 seconds, the ball lands at the point with position vector \((40\mathbf{i} - 10\mathbf{j})\) metres. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively. Assume that there are no resistance forces acting on the ball.

1. Find the speed of the ball when it is at a height of 3 metres above its initial position. [6 marks]
2. State the speed of the ball when it is at its maximum height. [1 mark]
3. Explain why the answer you found in part (b) may not be the actual speed of the ball when it is at its maximum height. [1 mark]

A particle P has position vector \(\mathbf{r}\) m at time \(t\) s given by \(\mathbf{r} = (t^3 - 3t^2)\mathbf{i} - (4t^2 + 1)\mathbf{j}\) for \(t \geq 0\). Find the magnitude of the acceleration of P when \(t = 2\). [4]

Two particles \(A\) and \(B\) have position vectors \(\mathbf{r}_A\) metres and \(\mathbf{r}_B\) metres at time \(t\) seconds, where $$\mathbf{r}_A = t^2\mathbf{i} + (3t - 1)\mathbf{j} \quad \text{and} \quad \mathbf{r}_B = (1 - 2t^2)\mathbf{i} + (3t - 2t^2)\mathbf{j}, \quad \text{for } t \geqslant 0.$$

1. Find the values of \(t\) when \(A\) and \(B\) are moving with the same speed. [5]
2. Show that the distance, \(d\) metres, between \(A\) and \(B\) at time \(t\) satisfies $$d^2 = 13t^4 - 10t^2 + 2.$$ [3]
3. Hence find the shortest distance between \(A\) and \(B\) in the subsequent motion. [6]

In this question you must show detailed reasoning. \includegraphics{figure_11} A football \(P\) is kicked with speed \(25\,\text{m}\,\text{s}^{-1}\) at an angle of elevation \(\alpha\) from a point \(A\) on horizontal ground. At the same instant a second football \(Q\) is kicked with speed \(15\,\text{m}\,\text{s}^{-1}\) at an angle of elevation \(2\alpha\) from a point \(B\) on the same horizontal ground, where \(AB = 72\) m. The footballs are modelled as particles moving freely under gravity in the same vertical plane and they collide with each other at the point \(C\) (see diagram).

1. Calculate the height of \(C\) above the ground. [7]
2. Find the direction of motion of \(P\) at the moment of impact. [4]
3. Suggest one improvement that could be made to the model. [1]

A particle \(P\) moves with constant acceleration \((3\mathbf{i} - 5\mathbf{j})\text{m s}^{-2}\). At time \(t = 0\) seconds \(P\) is at the origin. At time \(t = 4\) seconds \(P\) has velocity \((2\mathbf{i} + 4\mathbf{j})\text{m s}^{-1}\).

1. Find the displacement vector of \(P\) at time \(t = 4\) seconds. [2]
2. Find the speed of \(P\) at time \(t = 0\) seconds. [4]

In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. A particle of mass 0.12 kg is moving so that its position vector \(\mathbf{r}\) metres at time \(t\) seconds is given by \(\mathbf{r} = 2t^2\mathbf{i} + (5t^2 - 4t)\mathbf{j}\).

1. Show that when \(t = 0.7\) the bearing on which the particle is moving is approximately \(044°\). [3]
2. Find the magnitude of the resultant force acting on the particle at the instant when \(t = 0.7\). [4]
3. Determine the times at which the particle is moving on a bearing of \(045°\). [2]

A girl can paddle her canoe at \(5 \text{ m s}^{-1}\) in still water. She wishes to cross a river which is \(100 \text{ m}\) wide and flowing at \(8 \text{ m s}^{-1}\).

1. 1. Write down the angle to the river bank at which the boat must head, in order to cross the river in the least possible time. [1]
   2. Find the acute angle to the river bank at which the boat must head, in order to cross the river by the shortest route. [4]
2. Calculate the times taken for each of the two cases in part (i). [3]