Shout toward a cliff and you hear your voice come back: an echo, sound reflected off a surface. If you know the speed of sound and time the echo, you can measure distances you can't reach. Bats, dolphins, ships' sonar and ultrasound scanners all do this.
Key idea
$$d = \dfrac{v\,t}{2}$$ where $t$ is the time from sending the sound to hearing the echo. Divide by 2 because the sound travels there and back.
Why it works
In time $t$, the sound covers a total path of $v\,t$. That path goes out to the reflecting surface and returns, so it is twice the distance to the surface. Half of it, $\dfrac{v\,t}{2}$, is the one-way distance.
Worked example: Sonar
A ship sends a sound pulse straight down and hears the echo from the seabed 2 s later. Sound travels at 1,500 m/s in seawater. How deep is the sea?
- Total path: $1{,}500 \times 2 = 3{,}000$ m, down and back.
- Depth: $3{,}000 \div 2 = 1{,}500$ m.
Answer: 1,500 m
Worked example: A canyon wall
You clap and hear the echo from a canyon wall 1.2 s later. Sound travels at 343 m/s. How far away is the wall?
- $d = \dfrac{343 \times 1.2}{2} = \dfrac{411.6}{2} = 205.8$ m.
Answer: About 206 m
Common mistake
Forgetting that the sound travels there and back, which gives an answer twice the real distance. Check: if your answer is the whole distance the sound traveled in the time, you still need to halve it.
Many bats chirp at frequencies too high for us to hear and judge distances to insects from the echoes, within a fraction of a second.