Two spectators at a soccer game see, and a moment later...

Two spectators at a soccer game see, and a moment later hear, the ball being kicked on the playing field. The time delay for spectator $A$ is 0.23 s, and for spectator $B$ it is 0.12 s. Sight lines from the two spectators to the player kicking the ball meet at an angle of $90^{\circ} .$ How far are (a) spectator $A$ and $(b)$ spectator $B$ from the player? (c) How far are the spectators from each other?

Two spectators at a soccer game see, and a moment later hear, the ball being kicked on the playing field. The time delay for spectator $A$ is 0.23 s, and for spectator $B$ it is 0.12 s. Sight lines from the two spectators to the player kicking the ball meet at an angle of $90^{\circ} .$ How far are (a) spectator $A$ and $(b)$ spectator $B$ from the
player? (c) How far are the spectators from each other?

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Key Concepts

Speed of Sound

This concept pertains to how fast sound waves propagate through a medium such as air. In problems where there is a delay between an event and when it is heard, the speed of sound is used to convert this time delay into a distance. For instance, if you know the delay time and the speed, you can calculate how far away the event occurred.

Time Delay in Sound Propagation

This key concept involves understanding that the time difference between seeing an event and hearing it is due to the finite speed at which sound travels. By using the measured time delay and the speed of sound, you can determine the distance from the observer to the source of the sound.

Right Triangle Geometry

This concept is utilized when the lines of sight from different observers to a common point form known angles. When these sight-lines intersect at a right angle (90°), the distances from each observer to the event can be used as the perpendicular sides of a right triangle. The Pythagorean theorem then allows the computation of the distance between the observers, making it a practical application of basic geometric principles.

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