1.10c Magnitude and direction: of vectors

6 Two forces, \(\mathbf { P } = ( 6 \mathbf { i } - 3 \mathbf { j } )\) newtons and \(\mathbf { Q } = ( 3 \mathbf { i } + 15 \mathbf { j } )\) newtons, act on a particle. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular.

1. Find the resultant of \(\mathbf { P }\) and \(\mathbf { Q }\).
2. Calculate the magnitude of the resultant of \(\mathbf { P }\) and \(\mathbf { Q }\).
3. When these two forces act on the particle, it has an acceleration of \(( 1.5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Find the mass of the particle.
4. The particle was initially at rest at the origin.
   1. Find an expression for the position vector of the particle when the forces have been applied to the particle for \(t\) seconds.
   2. Find the distance of the particle from the origin when the forces have been applied to the particle for 2 seconds.

7 A boat is travelling in water that is moving north-east at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of the boat relative to the water is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due west. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Show that the magnitude of the resultant velocity of the boat is \(3.85 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
2. Find the bearing on which the boat is travelling, giving your answer to the nearest degree.

4 A canoe is paddled across a river which has a width of 20 metres. The canoe moves from the point \(X\) on one bank of the river to the point \(Y\) on the other bank, so that its path is a straight line at an angle \(\alpha\) to the banks. The velocity of the canoe relative to the water is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the banks. The water flows at \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to the banks. \includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-10_469_1333_543_374} Model the canoe as a particle.

1. Find the magnitude of the resultant velocity of the canoe.
2. Find the angle \(\alpha\).
3. Find the time that it takes for the canoe to travel from \(X\) to \(Y\).
   \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-11_2486_1714_221_153}

5 A particle moves with constant acceleration \(( - 0.4 \mathbf { i } + 0.2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Initially, it has velocity \(( 4 \mathbf { i } + 0.5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Find an expression for the velocity of the particle at time \(t\) seconds.
2. 1. Find the velocity of the particle when \(t = 22.5\).
   2. State the direction in which the particle is travelling at this time.
3. Find the time when the speed of the particle is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

4 A small ferry is used to cross a river which has straight parallel banks that are 200 metres apart. The water in the river moves at a constant speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ferry travels from a point \(A\) on one bank to a point \(B\) directly opposite \(A\) on the other bank. The velocity of the ferry relative to the water is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the upstream bank, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{d42b2e88-74ea-486b-bb47-f512eb0c185d-4_467_1122_550_459}

1. \(\quad\) Find \(V\).
2. Find the time that it takes for the ferry to cross the river, giving your answer to the nearest second.

7 A helicopter is initially at rest on the ground at the origin when it begins to accelerate in a vertical plane. Its acceleration is \(( 4.2 \mathbf { i } + 2.5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\) for the first 20 seconds of its motion. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical respectively. Assume that the helicopter moves over horizontal ground.

1. Find the height of the helicopter above the ground at the end of the 20 seconds.
2. Find the velocity of the helicopter at the end of the 20 seconds.
3. Find the speed of the helicopter when it is at a height of 180 metres above the ground.

2 Three forces act on a particle. These forces are ( \(9 \mathbf { i } - 3 \mathbf { j }\) ) newtons, ( \(5 \mathbf { i } + 8 \mathbf { j }\) ) newtons and ( \(- 7 \mathbf { i } + 3 \mathbf { j }\) ) newtons. The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.

1. Find the resultant of these forces.
2. Find the magnitude of the resultant force.
3. Given that the particle has mass 5 kg , find the magnitude of the acceleration of the particle.
4. Find the angle between the resultant force and the unit vector \(\mathbf { i }\).

6 A river has straight parallel banks. The water in the river is flowing at a constant velocity of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to the banks. A boat crosses the river, from the point \(A\) to the point \(B\), so that its path is at an angle \(\alpha\) to the bank. The velocity of the boat relative to the water is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the bank. The diagram shows these velocities and the path of the boat. \includegraphics[max width=\textwidth, alt={}, center]{ccc1db66-9700-4f22-905e-cc0bdf1fd3c1-12_467_988_568_532}

1. Show that \(\alpha = 53.1 ^ { \circ }\), correct to three significant figures.
2. The boat returns along the same straight path from \(B\) to \(A\). Given that the speed of the boat relative to the water is still \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the magnitude of the resultant velocity of the boat on the return journey.

7 A particle is initially at the point \(A\), which has position vector \(13.6 \mathbf { i }\) metres, with respect to an origin \(O\). At the point \(A\), the particle has velocity \(( 6 \mathbf { i } + 2.4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), and in its subsequent motion, it has a constant acceleration of \(( - 0.8 \mathbf { i } + 0.1 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Find an expression for the velocity of the particle \(t\) seconds after it leaves \(A\).
2. Find an expression for the position vector of the particle, with respect to the origin \(O\), \(t\) seconds after it leaves \(A\).
3. Find the distance of the particle from the origin \(O\) when it is travelling in a north-westerly direction.
   \includegraphics[max width=\textwidth, alt={}]{ccc1db66-9700-4f22-905e-cc0bdf1fd3c1-17_2486_1709_221_153}

1 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface when they collide. During the collision, the two particles coalesce to form a single combined particle. Particle \(A\) has mass 3 kg and particle \(B\) has mass 7 kg . Before the collision, the velocity of \(A\) is \(\left[ \begin{array} { r } 6 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { r } - 1 \\ 4 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the velocity of the combined particle after the collision.
2. Find the speed of the combined particle after the collision.

4 A river has parallel banks which are 16 metres apart. The water in the river flows at \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to the banks. A boat sets off from one bank at the point \(A\) and travels perpendicular to the bank so that it reaches the point \(B\), which is directly opposite the point \(A\). It takes the boat 10 seconds to cross the river. The velocity of the boat relative to the water has magnitude \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is at an angle \(\alpha\) to the bank, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-08_400_1011_667_511}

1. Show that the magnitude of the resultant velocity of the boat is \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. \(\quad\) Find \(V\).
3. Find \(\alpha\).
4. State one modelling assumption that you needed to make about the boat.

   	\includegraphics[max width=\textwidth, alt={}]{c022c936-72bc-4cf9-8f98-285f12c1d479-08_72_1689_1617_154}

   .......... \(\_\_\_\_\) \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-09_40_118_529_159}

5 An aeroplane is travelling along a straight line between two points, \(A\) and \(B\), which are at the same height. The air is moving due east at a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Relative to the air, the aeroplane travels due north at a speed of \(100 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the magnitude of the resultant velocity of the aeroplane.
   (3 marks)
2. Find the bearing on which the aeroplane is travelling, giving your answer to the nearest degree.
   (2 marks)
   

   QUESTION PART REFERENCE

   \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-11_2484_1709_223_153}

7 A helicopter is initially hovering above a lighthouse. It then sets off so that its acceleration is \(( 0.5 \mathbf { i } + 0.375 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). The helicopter does not change its height above sea level as it moves. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Find the speed of the helicopter 20 seconds after it leaves its position above the lighthouse.
2. Find the bearing on which the helicopter is travelling, giving your answer to the nearest degree.
3. The helicopter stops accelerating when it is 500 metres from its initial position. Find the time that it takes for the helicopter to travel from its initial position to the point where it stops accelerating.

1 As a boat moves, it travels at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due north, relative to the water. The water is moving due west at \(2 \mathrm {~ms} ^ { - 1 }\).

1. Find the magnitude of the resultant velocity of the boat.
2. Find the bearing of the resultant velocity of the boat.

3 A ship travels through water that is moving due east at a speed of \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ship travels due north relative to the water at a speed of \(7 \mathrm {~ms} ^ { - 1 }\). The resultant velocity of the ship is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a bearing \(\alpha\). \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. \(\quad\) Find \(V\).
2. Find \(\alpha\), giving your answer as a three-figure bearing, correct to the nearest degree.

4 Two forces, acting at a point, have magnitudes of 40 newtons and 70 newtons. The angle between the two forces is \(30 ^ { \circ }\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{cb5001b1-1744-439f-aa35-8dd01bc90421-3_213_531_400_759}

1. Find the magnitude of the resultant of these two forces.
2. Find the angle between the resultant force and the 70 newton force.

4 A boat is crossing a river, which has two parallel banks. The width of the river is 20 metres. The water in the river is flowing at a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The boat sets off from the point \(O\) on one bank. The point \(A\) is directly opposite \(O\) on the other bank. The velocity of the boat relative to the water is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(70 ^ { \circ }\) to the bank. The boat lands at the point \(B\) which is 3 metres from \(A\). The angle between the actual path of the boat and the bank is \(\alpha ^ { \circ }\). The river and the velocities are shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{788534a5-abbb-4d6a-87b2-c54e859a128a-10_490_1307_641_370}

1. Find the time that it takes for the boat to cross the river.
2. Find \(\alpha\).
3. \(\quad\) Find \(V\).

2 A yacht is sailing through water that is flowing due west at \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of the yacht relative to the water is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due south. The yacht has a resultant velocity of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a bearing of \(\theta\).

1. \(\quad\) Find \(V\).
2. Find \(\theta\), giving your answer to the nearest degree.

4 A particle moves with constant acceleration between the points \(A\) and \(B\). At \(A\), it has velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At \(B\), it has velocity \(( 7 \mathbf { i } + 6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). It takes 10 seconds to move from \(A\) to \(B\).

1. Find the acceleration of the particle.
2. Find the distance between \(A\) and \(B\).
3. Find the average velocity as the particle moves from \(A\) to \(B\).

4 An aeroplane is flying in air that is moving due east at \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Relative to the air, the aeroplane has a velocity of \(90 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due north. During a 20 second period, the motion of the air causes the aeroplane to move 240 metres to the east.

1. \(\quad\) Find \(V\).
2. Find the magnitude of the resultant velocity of the aeroplane.
3. Find the direction of the resultant velocity, giving your answer as a three-figure bearing, correct to the nearest degree.
   [0pt] [3 marks]

6. The points \(A\) and \(B\) have position vectors \(( 30 \mathbf { i } - 60 \mathbf { j } ) \mathrm { m }\) and \(( - 20 \mathbf { i } + 60 \mathbf { j } ) \mathrm { m }\) respectively relative to an origin \(O\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors. A cyclist, Chris, starts at \(A\) and cycles towards \(B\) with constant speed \(2.6 \mathrm {~ms} ^ { - 1 }\). Another cyclist, Doug, starts at \(O\) and cycles towards \(B\) with constant speed \(k \sqrt { } 10 \mathrm {~ms} ^ { - 1 }\).

1. Show that Chris's velocity vector is \(( - \mathbf { i } + 2 \cdot 4 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
2. Find Doug's velocity vector in the form \(k ( a \mathbf { i } + b \mathbf { j } ) \mathrm { ms } ^ { - 1 }\). Given that Chris and Doug arrive at \(B\) at the same time,
3. find the value of \(k\).

3. In a simple model for the motion of a car, its velocity, \(\mathbf { v }\), at time \(t\) seconds, is given by $$\mathbf { v } = \left( 3 t ^ { 2 } - 2 t + 8 \right) \mathbf { i } + ( 5 t + 6 ) \mathbf { j } \mathrm { ms } ^ { - 1 }$$

1. Calculate the speed of the car when \(t = 0\).
2. Find the values of \(t\) for which the velocity of the car is parallel to the vector \(( \mathbf { i } + \mathbf { j } )\).
3. Why would this model not be appropriate for large values of \(t\) ?

2. A ball of mass 2 kg moves on a smooth horizontal surface under the action of a constant force, \(\mathbf { F }\). The initial velocity of the ball is \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and 4 seconds later it has velocity \(( 10 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular, horizontal unit vectors.

1. Making reference to the mass of the ball and the force it experiences, explain why it is reasonable to assume that the acceleration is constant.
2. Find, giving your answers correct to 3 significant figures,
   1. the magnitude of the acceleration experienced by the ball,
   2. the angle which \(\mathbf { F }\) makes with the vector \(\mathbf { i }\).

2. A particle of mass 8 kg moves in a horizontal plane and is acted upon by three forces \(\mathbf { F } _ { 1 } = ( 5 \mathbf { i } - 3 \mathbf { j } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 3 } = ( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors.

1. Find the magnitude, in newtons, of the resultant force which acts on the particle, giving your answer in the form \(k \sqrt { } 5\).
2. Calculate, giving your answer in degrees correct to 1 decimal place, the angle the acceleration of the particle makes with the vector \(\mathbf { i }\).

4. The position of an aeroplane flying in a straight horizontal line at constant speed is plotted on a radar screen. At 2 p.m. the position vector of the aeroplane is \(( 80 \mathbf { i } + 5 \mathbf { j } )\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors directed east and north respectively relative to a fixed origin, \(O\), on the screen. Ten minutes later the position of the aeroplane on the screen is \(( 32 \mathbf { i } + 19 \mathbf { j } )\). Each unit on the screen represents 1 km .

1. Find the position vector of the aeroplane at 2:30 p.m.
2. Find the speed of the aeroplane in \(\mathrm { km } \mathrm { h } ^ { - 1 }\).
3. Find, correct to the nearest degree, the bearing on which the aeroplane is flying.