Relative velocity: find resultant velocity (magnitude and/or direction)

6 A motor boat can travel at a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the water. It is used to cross a river in which the current flows at \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resultant velocity of the boat makes an angle of \(60 ^ { \circ }\) to the river bank, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-4_561_1339_1692_350} The angle between the direction in which the boat is travelling relative to the water and the resultant velocity is \(\alpha\).

1. Show that \(\alpha = 16.8 ^ { \circ }\), correct to three significant figures.
2. Find the magnitude of the resultant velocity.

2 The velocity of a ship, relative to the water in which it is moving, is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due north. The water is moving due east with a speed of \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resultant velocity of the ship has magnitude \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find \(U\).
2. Find the direction of the resultant velocity of the ship. Give your answer as a bearing to the nearest degree.

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 An aeroplane is travelling due north at \(180 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the air. The air is moving north-west at \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the magnitude of the resultant velocity of the aeroplane.
2. Find the direction of the resultant velocity, giving your answer as a three-figure bearing to the nearest degree.

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}

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.

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 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]

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively.]

An aeroplane makes a journey from a point \(P\) to a point \(Q\) which is due east of \(P\). The wind velocity is \(w ( \cos \theta \mathbf { i } + \sin \theta \mathbf { j } )\), where \(w\) is a positive constant. The velocity of the aeroplane relative to the wind is \(v ( \cos \phi \mathbf { i } - \sin \phi \mathbf { j } )\), where \(v\) is a constant and \(v > w\). Given that \(\theta\) and \(\phi\) are both acute angles,

1. show that \(v \sin \phi = w \sin \theta\),
2. find, in terms of \(v , w\) and \(\theta\), the speed of the aeroplane relative to the ground.

6. Two particles \(P\) and \(Q\) have constant velocity vectors \(\mathbf { v } _ { P }\) and \(\mathbf { v } _ { Q }\) respectively. The magnitude of the velocity of \(P\) relative to \(Q\) is equal to the speed of \(P\). If the direction of motion of one of the particles is reversed, the magnitude of the velocity of \(P\) relative to \(Q\) is doubled. Find

1. the ratio of the speeds of \(P\) and \(Q\),
2. the cosine of the angle between the directions of motion of \(P\) and \(Q\).

5 An aeroplane flies in air that is moving due east at a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of the aeroplane relative to the air is \(150 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due north. The aeroplane actually travels on a bearing of \(030 ^ { \circ }\).

1. Show that \(V = 86.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
2. Find the magnitude of the resultant velocity of the aeroplane.

\includegraphics{figure_1}

1. \(A\) has velocity \(\vec{x}\) and \(C\) has velocity \(\vec{v}\)

\includegraphics{figure_2}

1. 1. \(C\)
   2. \(C\) has velocity \(\vec{v}\)

\(A\) has velocity \(\vec{v}\), there are velocities \(\vec{x}\), \(\vec{v}\), \(\vec{v}\) around point \(O\), and velocity \(\vec{v}\)

1. 1. —
   2. \(v\) and \(v\)

\(A\) has velocity \(\vec{v}\)

1. 1. \(C\) with velocity \(\vec{v}\)
   2. \(C\)

\includegraphics{figure_5} \(A\) has velocity \(\vec{x}\), there are velocities \(\vec{x}\), \(\vec{x}\), \(\vec{v}\) and \(\vec{v}\)

1. 1. \(v\)
   2. \(v\) and \(v\)

\includegraphics{figure_6} \(E\) has velocity \(\vec{v}\)

1. 1. \(B\)
   2. \(v\)

\(A\) has velocity \(\vec{x}\)

1. 1. \(C\)
2. \(A\) has velocity \(\vec{x}\)
   1. \(C\)
3. \(C\) with velocities \(v \vec{v}\)

Particle \(A\) has velocity \((8\mathbf{i} - 3\mathbf{j})\) ms\(^{-1}\) and particle \(B\) has velocity \((15\mathbf{i} - 8\mathbf{j})\) ms\(^{-1}\) where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular, horizontal unit vectors.

1. Find the speed of \(B\). [2 marks]
2. Find the velocity of \(B\) relative to \(A\). [2 marks]
3. Find the acute angle between the relative velocity found in part (b) and the vector \(\mathbf{i}\), giving your answer in degrees correct to 1 decimal place. [2 marks]