Two spectators at a soccer game in Montjuic Stadium see, and...

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 $0.23 \mathrm{~s}$, and for spectator $B$ it is $0.12 \mathrm{~s}$. 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?

Key Concepts

Right Triangle Geometry

Right triangle geometry involves using the properties of triangles with one 90° angle. In cases where the observers’ sight lines intersect at a right angle, the Pythagorean theorem becomes applicable to calculate unknown distances. This method simplifies the problem of finding the distance between two points and is essential in deriving the relationships between the distances involved.

Geometric Triangulation

Geometric triangulation is a technique used to determine the position or distance of an object based on angle measurements from two or more observation points. By knowing the angles at which observers detect the source and combining this with distance information, it is possible to form triangles that accurately represent the spatial relationships between the points involved.

Sound Propagation

This concept pertains to the way sound travels through a medium, such as air, at a finite speed. In many physics problems, including this one, the speed of sound is a key factor because it determines how long it takes for sound to travel a given distance. This idea is used to relate the time delay between when an event occurs and when it is heard to the distance the sound has traveled.

Time Delay Measurement

Time delay measurement involves recording the difference in time between the occurrence of an event and its detection by an observer. In scenarios where information (like sound) travels at a finite speed, these measurements are critical in calculating distances. In this problem, the delay between seeing the event and hearing it directly leads to determining the distance between the observer and the source.

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