A game requires that two children each throw a ball upward...

A game requires that two children each throw a ball upward as high as possible from point $O$ and then run horizontally in opposite directions away from O. The child who travels the greater distance before their thrown ball impacts the ground wins. If child $A$ throws a ball upward with a speed of $v_{1}=70 \mathrm{ft} / \mathrm{sec}$ and immediately runs leftward at a constant speed of $v_{A}=16$ ft/sec while child $B$ throws the ball upward with a speed of $v_{2}=64 \mathrm{ft} / \mathrm{sec}$ and immediately runs rightward with a constant speed of $v_{B}=18 \mathrm{ft} / \mathrm{sec},$ which child will win the game?

A game requires that two children each throw a ball upward as high as possible from point $O$ and then run horizontally in opposite directions away from
O. The child who travels the greater distance before their thrown ball impacts the ground wins. If child $A$ throws a ball upward with a speed of $v_{1}=70 \mathrm{ft} / \mathrm{sec}$ and immediately runs leftward at a constant speed of $v_{A}=16$ ft/sec while child $B$ throws the ball upward with a speed of $v_{2}=64 \mathrm{ft} / \mathrm{sec}$ and immediately runs rightward with a constant speed of $v_{B}=18 \mathrm{ft} / \mathrm{sec},$ which child will win the game?

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Key Concepts

Projectile Motion

Projectile motion describes the two-dimensional movement of an object that is launched into the air and is affected only by gravity (assuming air resistance is negligible). It involves an upward motion decelerated by gravity until the object stops momentarily at the peak, followed by a downward acceleration. This concept is crucial in determining how long an object remains airborne.

Time of Flight Calculation

The time of flight refers to the total duration an object, such as a ball thrown upward, remains in the air before landing. This can be calculated by using the object's initial vertical speed and the acceleration due to gravity. Because of the symmetrical nature of projectile motion, the total time is twice the time taken to reach the maximum height.

Horizontal Motion Analysis

Horizontal motion analysis involves calculating the distance traveled along the horizontal direction when an object or person moves at a constant speed. In scenarios where the movement is separate from the vertical projectile motion, the horizontal distance can be determined by multiplying the constant speed by the time duration for which the motion occurs. This is key in comparing distances when different speeds or time intervals are involved.

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