Two spectators at a soccer game in Montjuic Stadium see, and a moment later hear, the ball being kicked on the playing field. The time delay for spectator A is 0.201 s and for spectator B is 0.122 s. Sight lines from the two spectators to the player kicking the ball meet at an angle of 90°. How far are (a) spectator A and (b) spectator B from the player? (c) How far are the spectators from each other? (Take the speed of sound to be 343 m/s.)

a) 68.943 m

b) 41.846 m

c) 80.648 m

Explanation:

Given:

Delay in time for spectator A, t₁ = 0.201 s

Delay in time for spectator B, t₂ = 0.122 s

the delay in sound heard is the due to the distance being traveled by the sound from the kicker to the spectator

thus,

a) Distance of the kicker from A,

d₁ = speed of sound * time taken

d₁ = 343 m/s * 0.201 s = 68.943 m

b) Distance of the kicker from B,

d₂ = speed of sound * time taken

d₂ = 343 m/s * 0.122 = 41.846 m

c) Since the angle between the two spectators for the player is 90°

thus, a right angles triangle is formed.

where, the distance between the spectators is the hypotenuses (s) of the so formed triangle

Therefore,

s² = d₁² + d₂²

on substituting the values, we get

s² = 68.943² + 41.846²

or

s² = 6504.22

or

s = √6504.22

or

s = 80.648 m

hence, the distance between the spectators is 80.648 m