Chapter 4: Review

The chapter problem, solved

A survey boat checks whether a harbour channel is deep enough for a ferry. The speed of sound in sea water is 1500 m/s. A pulse returns from the sea floor after 0.020 s. The echo sounder works at 50 kHz. The ripple tank makes waves at 5.0 Hz with a wavelength of 0.12 m.

1. The depth of the sea floor. The pulse travels to the sea floor and back, so the depth is half of the distance the pulse covers.

\begin{aligned} s &= \frac{v t}{2} \\ &= \frac{1500 \times 0.020}{2} \\ &= \frac{30}{2} \\ &= 15\,\text{m} \end{aligned}

The water is 15 m deep. The ferry needs at least 12 m, so the channel is deep enough.

2. The wavelength of the echo sounder. The frequency is 50 kHz, which is 50000 Hz.

\begin{aligned} v &= f \lambda \\ \lambda &= \frac{v}{f} \\ &= \frac{1500}{50000} \\ &= 0.030\,\text{m} \end{aligned}

The wavelength is 0.030 m, which is 3.0 cm.

3. The speed of the ripples.

\begin{aligned} v &= f \lambda \\ &= 5.0 \times 0.12 \\ &= 0.60\,\text{m/s} \end{aligned}

4. The period of the ripples.

T = \frac{1}{f} = \frac{1}{5.0} = 0.20\,\text{s}

5. The reflection at the sea floor. When the sound pulse reaches the sea floor, the boundary between water and sediment, part of the pulse is reflected back to the boat. The rest is transmitted into the sediment or absorbed by it. The reflected part is the echo the boat times, so the echo gives the depth.

6. A medical scanner. The scanner uses ultrasound at 2.0 MHz, which is $2.0 \times 10^6\,\text{Hz}$ .

\begin{aligned} \lambda &= \frac{v}{f} \\ &= \frac{1500}{2.0 \times 10^6} \\ &= 7.5 \times 10^{-4}\,\text{m} \end{aligned}

This is 0.75 mm. Shorter wavelengths can show smaller details.

7. A seismic wave. A P-wave covers 60 km in 12 s.

\begin{aligned} v &= \frac{s}{t} \\ &= \frac{60000}{12} \\ &= 5000\,\text{m/s} \end{aligned}

P-waves and S-waves give evidence about the size and structure of the Earth's core, which cannot be seen directly.

The judgement that the channel is deep enough is a reasoned answer. It follows from the numbers above. Whether a plan is safe is not the kind of claim a computer can check, so compare your reasoning with the evidence in the working.

Mixed practice — choose the method first

For each question, say which equation or idea you will use before you calculate. Keep your units with your numbers.

In the chapter problem a survey boat sends a sound pulse down at 1500 m/s. The reflection returns after 0.020 s. Calculate the depth of the sea floor.
The echo sounder in the chapter problem sends sound at a frequency of 50 kHz. The speed of sound in sea water is 1500 m/s. Calculate the wavelength.
In the chapter problem a ripple tank makes waves with a frequency of 5.0 Hz and a wavelength of 0.12 m. Calculate the wave speed.
The ripple tank in the chapter problem has a frequency of 5.0 Hz. Calculate the period of the waves.
H A medical scanner uses ultrasound at a frequency of 2.0 MHz. The speed of sound in tissue is 1500 m/s. Calculate the wavelength.
H A seismic P-wave travels 60 km in 12 s. Calculate its speed.
A ray of light meets a mirror at an angle of incidence of 35 degrees. State the angle of reflection, and say what line the angles are measured from.
Describe what happens to a wave at the boundary between two different materials.
A train travels 360 m in 20 s. Calculate its average speed.
A ball of mass 2.0 kg moves at 3.0 m/s. Calculate the energy in its kinetic store.
A pupil has a mass of 50 kg. Take g = 9.8 N/kg. Calculate the weight of the pupil.
A resultant force acts on a trolley of mass 800 kg and gives it an acceleration of 2.0 m/s squared. Calculate the resultant force.
A force of 60 N moves a box 4.0 m along the line of the force. Calculate the work done.
A sound wave passes from air into water. Say which ideas you would use to find the wavelength in the water, and name the quantities you need.
H A pulse of ultrasound travels through tissue at 1500 m/s. The reflection returns after 0.00016 s. Calculate the distance to the boundary.
H Compare the P-waves and the S-waves produced by an earthquake, and explain how they give evidence about the Earth.

Answers and concise solutions

1. Depth of the sea floor

Answer: The pulse travels down and back, so $s = \dfrac{v t}{2} = \dfrac{1500 \times 0.020}{2} = 15\,\text{m}$ . The channel is deeper than the 12 m the ferry needs.

2. Wavelength of the echo sounder

Answer: $50\,\text{kHz} = 50000\,\text{Hz}$ , so $\lambda = \dfrac{v}{f} = \dfrac{1500}{50000} = 0.030\,\text{m}$ , which is 3.0 cm.

3. Speed of the ripples

Answer: $v = f\lambda = 5.0 \times 0.12 = 0.60\,\text{m/s}$ .

4. Period of the ripples

Answer: $T = \dfrac{1}{f} = \dfrac{1}{5.0} = 0.20\,\text{s}$ .

5. Wavelength of the ultrasound

Answer: $2.0\,\text{MHz} = 2.0 \times 10^6\,\text{Hz}$ , so $\lambda = \dfrac{v}{f} = \dfrac{1500}{2.0 \times 10^6} = 7.5 \times 10^{-4}\,\text{m}$ , which is 0.75 mm.

6. Speed of the P-wave

Answer: $60\,\text{km} = 60000\,\text{m}$ , so $v = \dfrac{s}{t} = \dfrac{60000}{12} = 5000\,\text{m/s} = 5.0 \times 10^3\,\text{m/s}$ .

7. The angle of reflection

The angle of reflection is 35 degrees, because the angle of reflection equals the angle of incidence.
Both angles are measured from the normal, which is the line at right angles to the mirror at the point where the ray hits it.

8. A wave at a boundary

Part of the wave can be reflected back from the boundary.
Part of the wave can be transmitted through the boundary into the second material. The speed of the wave may change when it is transmitted.
Part of the wave can be absorbed by the material, so its energy is transferred to the material.

9. Average speed of the train

Answer: $v = \dfrac{s}{t} = \dfrac{360}{20} = 18\,\text{m/s}$ .

10. Kinetic energy of the ball

Answer: $E_k = \tfrac{1}{2}mv^2 = 0.5 \times 2.0 \times 3.0^2 = 9.0\,\text{J}$ .

11. Weight of the pupil

Answer: $W = mg = 50 \times 9.8 = 490\,\text{N} = 4.9 \times 10^2\,\text{N}$ .

12. Resultant force on the trolley

Answer: $F = ma = 800 \times 2.0 = 1600\,\text{N} = 1.6 \times 10^3\,\text{N}$ .

13. Work done on the box

Answer: $W = Fs = 60 \times 4.0 = 240\,\text{J} = 2.4 \times 10^2\,\text{J}$ .

14. Choosing the method

Use v = f lambda, rearranged to lambda = v / f.
The frequency does not change when the wave crosses the boundary. It is set by the source.
I need the speed of sound in water, and the frequency. Then the wavelength is the speed divided by the frequency.

15. Distance to the boundary

Answer: $s = \dfrac{v t}{2} = \dfrac{1500 \times 0.00016}{2} = 0.12\,\text{m}$ . The boundary is 0.12 m away.

16. P-waves, S-waves and the Earth

A P-wave is longitudinal and an S-wave is transverse. Both travel out from the earthquake.
P-waves travel at different speeds through solids and liquids. S-waves cannot travel through a liquid.
S-waves do not pass through the liquid outer core, so a region on the far side of the Earth receives no S-waves. This is evidence for the size and structure of the core.

What you can now do

describe a wave in terms of amplitude, wavelength, frequency and period, and give the unit of each;
use the wave equation $v = f\lambda$ and the period equation $T = \dfrac{1}{f}$ ;
tell a transverse wave from a longitudinal wave, with an example of each;
recall the speed of sound in air, and explain why sound needs a medium;
describe a method to measure the speed of sound in air and the speed of ripples on water;
describe reflection, transmission and absorption at a material interface;
state the law of reflection and draw a ray diagram to show it;
explain what happens to the speed, frequency and wavelength when a wave is transmitted into a new material;
calculate a distance from an echo, and explain how ultrasound and seismic waves are used for detection and exploration (Higher tier).

Your relevance reflection

Return to the prompt from the chapter opener: where in your own week do you meet a wave? Write one more example now that you have finished the chapter. Has the chapter changed what you notice about a sound, a mirror or an echo?