A boy can throw a ball a maximum horizontal distance of R on...

A boy can throw a ball a maximum horizontal distance of $R$ on a level field. How far can he throw the same ball vertically upward? Assume that his muscles give the ball the same speed in each case.

Key Concepts

Projectile Motion

Projectile motion examines the trajectory of an object under the influence of gravity after it is thrown. The motion is typically separated into horizontal and vertical components, each of which is analyzed using kinematic equations. This concept is fundamental to understanding how objects travel through the air, whether thrown horizontally, vertically, or at an angle.

Optimal Launch Angle

For a given initial speed, the maximum horizontal distance (or range) is achieved when the object is thrown at a 45-degree angle. This angle equally distributes the initial velocity into vertical and horizontal components, maximizing the distance traveled on level ground.

Kinematic Equations

Kinematic equations describe the motion of an object under constant acceleration, such as gravity. They are used to calculate various aspects of projectile motion, including time of flight, maximum height, and range. In this context, knowing the equations allows one to deduce that the vertical and horizontal motions, while connected, yield different outcomes for identical initial speeds.

Comparison of Vertical and Horizontal Throws

When the same initial speed is used in a vertical throw and a horizontal projectile throw, the energy is distributed differently. In vertical throw, the ball converts kinetic energy entirely into vertical potential energy, reaching a maximum height of v²/(2g). In contrast, the optimal horizontal throw at 45° uses its energy to maximize distance, with a range of v²/g. This shows that the vertical throw achieves only half the distance (or height) relative to the horizontal range under optimal conditions.

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