During a baseball practice session, the cutoff man A...

During a baseball practice session, the cutoff man A executes a throw to the third baseman $B$. If the initial speed of the baseball is $v_{0}=130 \mathrm{ft} / \mathrm{sec},$ what launch angle $\theta$ is best if the ball is to arrive at third base at essentially ground level?

Key Concepts

Range Equation

The range equation expresses the total horizontal distance traveled by a projectile launched from and landing at the same height. It is given by R = (v0² sin(2?))/g, where v0 is the initial speed, ? is the launch angle, and g is the acceleration due to gravity. This formula is essential for determining how the launch angle affects the distance traveled.

Optimal Launch Angle

For a projectile launched and landing at the same elevation, the maximum horizontal distance is achieved when sin(2?) is maximized. Since the maximum value of the sine function is 1, this occurs at 2? = 90°, hence ? = 45°. This optimal angle is a key result in projectile motion that informs decisions about the best angle to achieve the desired horizontal reach.

Projectile Motion

This concept describes the motion of any object that is launched into the air and moves under the influence of gravity alone, neglecting air resistance. It involves breaking the motion into horizontal and vertical components, where the horizontal motion proceeds at a constant velocity while the vertical motion is uniformly accelerated due to gravity.

Kinematic Equations

These are the fundamental equations that relate displacement, velocity, acceleration, and time for objects in motion. In projectile motion, they are used separately in the horizontal (constant velocity) and vertical (constant acceleration) directions to determine the trajectory, time of flight, and landing position.

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