1.10h Vectors in kinematics: uniform acceleration in vector form

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively. Position vectors are given with respect to a fixed origin \(O\).] A ship \(S\) is moving with constant velocity \((3\mathbf{i} + 3\mathbf{j})\) km h\(^{-1}\). At time \(t = 0\), the position vector of \(S\) is \((-4\mathbf{i} + 2\mathbf{j})\) km.

1. Find the position vector of \(S\) at time \(t\) hours. [2]

A ship \(T\) is moving with constant velocity \((-2\mathbf{i} + n\mathbf{j})\) km h\(^{-1}\). At time \(t = 0\), the position vector of \(T\) is \((6\mathbf{i} + \mathbf{j})\) km. The two ships meet at the point \(P\).

1. Find the value of \(n\). [5]
2. Find the distance \(OP\). [4]

[In this question, the horizontal unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are directed due East and North respectively.] A coastguard station \(O\) monitors the movements of ships in a channel. At noon, the station's radar records two ships moving with constant speed. Ship \(A\) is at the point with position vector \((-5\mathbf{i} + 10\mathbf{j})\) km relative to \(O\) and has velocity \((2\mathbf{i} + 2\mathbf{j})\) km h\(^{-1}\). Ship \(B\) is at the point with position vector \((3\mathbf{i} + 4\mathbf{j})\) km and has velocity \((-2\mathbf{i} + 5\mathbf{j})\) km h\(^{-1}\).

1. Given that the two ships maintain these velocities, show that they collide. [6]

The coast guard radios ship \(A\) and orders it to reduce its speed to move with velocity \((\mathbf{i} + \mathbf{j})\) km h\(^{-1}\). Given that \(A\) obeys this order and maintains this new constant velocity,

1. find an expression for the vector \(\overrightarrow{AB}\) at time \(t\) hours after noon. [2]
2. find, to 3 significant figures, the distance between \(A\) and \(B\) at 1400 hours, [3]
3. Find the time at which \(B\) will be due north of \(A\). [2]

A particle \(P\) of mass 3 kg is moving under the action of a constant force \(\mathbf{F}\) newtons. At \(t = 0\), \(P\) has velocity \((3\mathbf{i} - 5\mathbf{j})\) m s\(^{-1}\). At \(t = 4\) s, the velocity of \(P\) is \((-5\mathbf{i} + 11\mathbf{j})\) m s\(^{-1}\). Find

1. the acceleration of \(P\), in terms of \(\mathbf{i}\) and \(\mathbf{j}\). [2]
2. the magnitude of \(\mathbf{F}\). [4]

At \(t = 6\) s, \(P\) is at the point \(A\) with position vector \((6\mathbf{i} - 29\mathbf{j})\) m relative to a fixed origin \(O\). At this instant the force \(\mathbf{F}\) newtons is removed and \(P\) then moves with constant velocity. Three seconds after the force has been removed, \(P\) is at the point \(B\).

1. Calculate the distance of \(B\) from \(O\). [6]

A particle \(P\) moves in a horizontal plane. The acceleration of \(P\) is \((-\mathbf{i} + 2\mathbf{j}) \text{ m s}^{-2}\). At time \(t = 0\), the velocity of \(P\) is \((2\mathbf{i} - 3\mathbf{j}) \text{ m s}^{-1}\).

1. Find, to the nearest degree, the angle between the vector \(\mathbf{j}\) and the direction of motion of \(P\) when \(t = 0\). [3]

At time \(t\) seconds, the velocity of \(P\) is \(\mathbf{v} \text{ m s}^{-1}\). Find

1. an expression for \(\mathbf{v}\) in terms of \(t\), in the form \(a\mathbf{i} + b\mathbf{j}\), [2]
2. the speed of \(P\) when \(t = 3\), [3]
3. the time when \(P\) is moving parallel to \(\mathbf{i}\). [2]

Two cars \(A\) and \(B\) are moving on straight horizontal roads with constant velocities. The velocity of \(A\) is \(20 \text{ m s}^{-1}\) due east, and the velocity of \(B\) is \((10\mathbf{i} + 10\mathbf{j}) \text{ m s}^{-1}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors directed due east and due north respectively. Initially \(A\) is at the fixed origin \(O\), and the position vector of \(B\) is \(300\mathbf{j}\) m relative to \(O\). At time \(t\) seconds, the position vectors of \(A\) and \(B\) are \(\mathbf{r}\) metres and \(\mathbf{s}\) metres respectively.

1. Find expressions for \(\mathbf{r}\) and \(\mathbf{s}\) in terms of \(t\). [3]
2. Hence write down an expression for \(\overrightarrow{AB}\) in terms of \(t\). [1]
3. Find the time when the bearing of \(B\) from \(A\) is \(045°\). [5]
4. Find the time when the cars are again 300 m apart. [6]

At time \(t\) seconds, \(t \geq 0\), a particle \(P\) has velocity \(\mathbf{v}\) m s\(^{-1}\), where $$\mathbf{v} = (27 - 3t^2)\mathbf{i} + (8 - t^3)\mathbf{j}$$ When \(t = 1\), the particle \(P\) is at the point with position vector \(\mathbf{r}\) m relative to a fixed origin \(O\), where \(\mathbf{r} = -5\mathbf{i} + 2\mathbf{j}\) Find

1. the magnitude of the acceleration of \(P\) at the instant when it is moving in the direction of the vector \(\mathbf{i}\), [5]
2. the position vector of \(P\) at the instant when \(t = 3\) [5]

At time \(t\) seconds, a particle \(P\) has position vector \(r\) metres relative to a fixed origin \(O\), where $$r = (t^2 + 2t)\mathbf{i} + (t - 2t^2)\mathbf{j}.$$ Show that the acceleration of \(P\) is constant and find its magnitude. [5]

A particle \(P\) of mass \(0.3\) kg is moving under the action of a single force \(F\) newtons. At time \(t\) seconds the velocity of \(P\), v m s\(^{-1}\), is given by $$\mathbf{v} = 3t^2\mathbf{i} + (6t - 4)\mathbf{j}.$$

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 \(P\) 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]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in a horizontal and upward vertical direction respectively] A particle \(P\) is projected from a fixed point \(O\) on horizontal ground with velocity \(u(\mathbf{i} + c\mathbf{j}) \text{ ms}^{-1}\), where \(c\) and \(u\) are positive constants. The particle moves freely under gravity until it strikes the ground at \(A\), where it immediately comes to rest. Relative to \(O\), the position vector of a point on the path of \(P\) is \((x\mathbf{i} + y\mathbf{j})\) m.

1. Show that $$y = cx - \frac{4.9x^2}{u^2}.$$ [5]

Given that \(u = 7\), \(OA = R\) m and the maximum vertical height of \(P\) above the ground is \(H\) m,

1. using the result in part (a), or otherwise, find, in terms of \(c\),
   1. \(R\)
   2. \(H\).
   [6]

Given also that when \(P\) is at the point \(Q\), the velocity of \(P\) is at right angles to its initial velocity,

1. find, in terms of \(c\), the value of \(x\) at \(Q\). [6]

At time \(t\) seconds, a particle \(P\) has position vector \(r\) metres relative to a fixed origin \(O\), where $$\mathbf{r} = (t^2 + 2t)\mathbf{i} + (t - 2t^2)\mathbf{j}.$$ Show that the acceleration of \(P\) is constant and find its magnitude. [5]

The velocity \(v\) m s\(^{-1}\) of a particle \(P\) at time \(t\) seconds is given by $$\mathbf{v} = (3t - 2)\mathbf{i} - 5t\mathbf{j}.$$

1. Show that the acceleration of \(P\) is constant. [2]

At \(t = 0\), the position vector of \(P\) relative to a fixed origin O is \(3\mathbf{i}\) m.

1. Find the distance of \(P\) from O when \(t = 2\). [6]

A cricket ball of mass 0.5 kg is struck by a bat. Immediately before being struck, the velocity of the ball is \((-30\mathbf{i})\) m s\(^{-1}\). Immediately after being struck, the velocity of the ball is \((16\mathbf{i} + 20\mathbf{j})\) m s\(^{-1}\).

1. Find the magnitude of the impulse exerted on the ball by the bat. [4]

In the subsequent motion, the position vector of the ball is \(\mathbf{r}\) metres at time \(t\) seconds. In a model of the situation, it is assumed that \(\mathbf{r} = [16t\mathbf{i} + (20t - 5t^2)\mathbf{j}]\). Using this model,

1. find the speed of the ball when \(t = 3\). [4]

A bee flies in a straight line from \(A\) to \(B\), where \(\overrightarrow{AB} = (3\mathbf{i} - 12\mathbf{j})\) m, in 5 seconds at a constant speed. Find

1. the straight-line distance \(AB\), [2 marks]
2. the speed of the bee, [2 marks]
3. the velocity vector of the bee. [2 marks]

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]

A particle \(P\), of mass \(2.5\) kg, initially at rest at the point \(O\), moves on a smooth horizontal surface with constant acceleration \((\mathbf{i} + 2\mathbf{j})\) ms\(^{-2}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in the directions due East and due North respectively. Find

1. the velocity vector of \(P\) at time \(t\) seconds after it leaves \(O\), \hfill [2 marks]
2. the magnitude and direction of the velocity of \(P\) when \(t = 7\), \hfill [3 marks]
3. the magnitude, in N, of the force acting on \(P\). \hfill [2 marks]

Two trains \(S\) and \(T\) are moving with constant speeds on straight tracks which intersect at the point \(O\). At 9.00 a.m. \(S\) has position vector \((-10\mathbf{i} + 24\mathbf{j})\) km and \(T\) has position vector \(25\mathbf{j}\) km relative to \(O\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in the directions due east and due north respectively. \(S\) is moving with speed 52 km h\(^{-1}\) and \(T\) is moving with speed 50 km h\(^{-1}\), both towards \(O\).

1. Show that the velocity vector of \(S\) is \((20\mathbf{i} - 48\mathbf{j})\) km h\(^{-1}\) and find the velocity vector of \(T\). \hfill [5 marks]
2. Find expressions for the position vectors of \(S\) and \(T\) at time \(t\) minutes after 9.00 a.m. \hfill [5 marks]
3. Show that the bearing of \(T\) from \(S\) remains constant during the motion, and find this bearing. \hfill [5 marks]
4. Show that if the trains continue at the given speeds they will collide. \hfill [2 marks]

A particle has mass 6 kg. A single force \((24e^{-2t}\mathbf{i} - 12t^3\mathbf{j})\) newtons acts on the particle at time \(t\) seconds. No other forces act on the particle.

1. Find the acceleration of the particle at time \(t\). [2 marks]
2. At time \(t = 0\), the velocity of the particle is \((-7\mathbf{i} - 4\mathbf{j}) \text{ m s}^{-1}\). Find the velocity of the particle at time \(t\). [4 marks]
3. Find the speed of the particle when \(t = 0.5\). [4 marks]

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 velocity of the particle, \(\mathbf{v} \text{ m s}^{-1}\), is given by $$\mathbf{v} = (8t - t^4)\mathbf{i} + 6e^{-3t}\mathbf{j}$$

1. Find an expression for the acceleration of the particle at time \(t\). [2 marks]
2. The mass of the particle is \(2\) kg.
   1. Find an expression for the force \(\mathbf{F}\) acting on the particle at time \(t\). [2 marks]
   2. Find the magnitude of \(\mathbf{F}\) when \(t = 1\). [3 marks]
3. Find the value of \(t\) when \(\mathbf{F}\) acts due south. [2 marks]
4. When \(t = 0\), the particle is at the point with position vector \((3\mathbf{i} - 5\mathbf{j})\) metres. Find an expression for the position vector, \(\mathbf{r}\) metres, of the particle at time \(t\). [4 marks]

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 particle \(P\) of mass 3 kg has position vector \(\mathbf{r} = (2t^2 - 4t)\mathbf{i} + (1 - t^2)\mathbf{j}\) m at time \(t\) seconds.

1. Find the velocity vector of \(P\) when \(t = 3\). [3 marks]
2. Find the magnitude of the force acting on \(P\), showing that this force is constant. [4 marks]

A particle \(P\) moves in a plane such that its position vector \(\mathbf{r}\) metres at time \(t\) seconds, relative to a fixed origin \(O\), is \(\mathbf{r} = t^2\mathbf{i} - 2t\mathbf{j}\).

1. Find the velocity vector of \(P\) at time \(t\) seconds. [2 marks]
2. Show that the direction of the acceleration of \(P\) is constant. [2 marks]
3. Find the value of \(t\) when the acceleration of \(P\) has magnitude 12 ms\(^{-2}\). [3 marks]

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]