Moving objects and a river's width

One height, two moments

A boat sails towards a cliff. An aeroplane flies level. In each case one thing never changes: the height. The cliff is always the same height above the sea, and the aeroplane stays at the same height above the ground.

At two moments, the horizontal distance is different, and so is the angle. That gives two triangles with the same height and two different bases.

• The distance at a moment is $h\cot\theta$.
• The distance moved is the difference: $h\cot\beta - h\cot\alpha$.
• The speed is $\dfrac{\text{distance moved}}{\text{time}}$.

Example 1: a balloon that drifts

A balloon drifts at a fixed height. From a point on the ground its angle of elevation is $60°$. After $12$ seconds it is $30°$. In that time the balloon has moved $240$ m. Find its height and its speed.

1. Two distances. At the first moment, $x_1 = h\cot 60° = \dfrac{h}{\sqrt{3}}$. At the second, $x_2 = h\cot 30° = h\sqrt{3}$.
2. The distance moved. $h\sqrt{3} - \dfrac{h}{\sqrt{3}} = 240$, so $\dfrac{2h}{\sqrt{3}} = 240$.
3. The height. $h = 120\sqrt{3} \approx 207.85$ m.
4. The speed. $\dfrac{240}{12} = 20$ m/s.

Example 2: a boat and a cliff

From the top of a cliff $75$ m high, the angle of depression of a boat is $30°$. The boat sails straight towards the foot of the cliff. After $20$ seconds, the angle of depression is $60°$. Find the speed of the boat and the time it still needs to reach the cliff.

1. Depression is elevation from the boat. Each angle of depression equals the angle of elevation of the top of the cliff from the boat.
2. The first distance. $x_1 = 75\cot 30° = 75\sqrt{3} \approx 129.90$ m.
3. The second distance. $x_2 = 75\cot 60° = \dfrac{75}{\sqrt{3}} \approx 43.30$ m.
4. The distance moved. $129.90 - 43.30 = 86.60$ m in $20$ s, so the speed is $\dfrac{86.60}{20} \approx 4.33$ m/s.
5. The time left. $\dfrac{43.30}{4.33} = 10$ s.

Example 3: the width of a river

A building $28$ m tall stands back from a river. From its top, the angle of depression of the near bank is $60°$ and of the far bank is $30°$. How wide is the river?

1. The near bank. $x_1 = 28\cot 60° = \dfrac{28}{\sqrt{3}} \approx 16.17$ m from the foot of the building.
2. The far bank. $x_2 = 28\cot 30° = 28\sqrt{3} \approx 48.50$ m from the foot.
3. The width. Both distances are measured from the same foot along one line, so we subtract: $48.50 - 16.17 = 32.33$ m.

Example 4: a shadow that grows

A pole stands on level ground. Its shadow is $18$ m longer when the angle of elevation of the Sun is $30°$ than when it is $45°$. How tall is the pole?

1. Two shadows. At $45°$, the shadow is $h\cot 45° = h$. At $30°$, it is $h\cot 30° = h\sqrt{3}$.
2. The difference. $h\sqrt{3} - h = 18$, so $h(\sqrt{3} - 1) = 18$.
3. Solve. $h = \dfrac{18}{0.732} \approx 24.59$ m.

Check. The two shadows are $24.59$ m and $24.59 \times 1.732 = 42.59$ m, which differ by $18$ m.

Practise with new numbers

Try it in Python

The program finds the distance a boat moves between two sightings, its speed, and the time it needs to reach the cliff.

Common mistakes

• Adding the two distances when the object moves along one line. Subtract the nearer from the farther.
• Forgetting the time. The speed needs the distance moved and the time taken.
• Measuring the angle of depression from the vertical. Measure it from the horizontal line through the observer.
• Leaving out the eye height. If the eyes are $1.5$ m above the ground, add $1.5$ m at the end.

Remember these

• At each moment the distance is $h\cot\theta$, and the height $h$ is the same.
• The distance moved is the difference of the two distances.
• Speed is the distance moved divided by the time taken.
• A river's width is the difference of the distances to its two banks.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. In a problem with a moving aeroplane, which quantity is the same at both sightings?

2. The angle of depression of a boat from a cliff is $40°$. What is the angle of elevation of the top of the cliff from the boat?

3. A boat covers $25$ m in $10$ s. What is its speed in metres per second, and how far does it go in one minute?

4. Why is a shadow longer when the Sun is lower in the sky?

5. Why do we subtract the two distances to find the width of a river in example 3?