A motorist travelling at a constant speed of 20\text{ m s}^{-1} directly towards a tall ve… — JAMB UTME

A motorist travelling at a constant speed of $20\text{ m s}^{-1}$ directly towards a tall vertical cliff sounds a horn. If the motorist hears the echo of the horn $2.0\text{ s}$ later and the speed of sound in air is $340\text{ m s}^{-1}$ , what was the distance of the car from the cliff at the moment the horn was sounded?

Answer: 360 m

Answer

The distance of the car from the cliff at the instant the horn was sounded was

360\text{ m}

When the motorist sounds the horn at an initial distance

D

from the cliff, the sound wave travels toward the cliff. In the

2.0\text{ s}

it takes for the echo to return, the car advances

40\text{ m}

toward the cliff (

20\text{ m s}^{-1} \times 2.0\text{ s}

). The returning echo meets the motorist at a distance of

(D - 40)\text{ m}

from the cliff. Consequently, the sound covers a total distance of

D + (D - 40) = 2D - 40\text{ m}

. Because the sound wave travels at

340\text{ m s}^{-1}

for

2.0\text{ s}

, the actual distance covered by sound is

340 \times 2.0 = 680\text{ m}

. Setting

2D - 40 = 680

gives

2D = 720\text{ m}

, which yields

D = 360\text{ m}

Step-by-Step Solution

Calculate the distance covered by the car while moving toward the cliff during the echo time interval.

d_{\text{car}} = 20\text{ m s}^{-1} \times 2.0\text{ s} = 40\text{ m}

The car continues to move closer to the cliff for the entire

2.0\text{ s}

period.

Set up an expression for the total distance covered by the sound wave.

d_{\text{sound}} = D + (D - 40) = 2D - 40\text{ m}

The sound travels forward a distance

D

to the cliff and reflects back to the car's updated location, which is

(D - 40)\text{ m}

from the cliff.

Calculate the distance travelled by sound using the given speed of sound.

d_{\text{sound}} = 340\text{ m s}^{-1} \times 2.0\text{ s} = 680\text{ m}

Sound propagates through air at

340\text{ m s}^{-1}

Equate the geometric path expression to the physical sound distance and solve for

D

2D - 40 = 680 \implies 2D = 720 \implies D = 360\text{ m}

Solving the equation yields the initial position of the car relative to the cliff.

Key Concept

Echo distance calculation with a moving observer

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