Question

A pitcher throws a baseball at \(40 \mathrm{~m} / \mathrm{s}(\sim 90 \mathrm{mph})\), and the catcher stops it in her glove. Is the work done on the ball by the catcher positive, negative, or zero? Explain.

Short answer

The work done is negative because the force by the catcher is opposite to the ball's motion.

Step-by-step solution

Understand the Context

Recall that work is done on an object when a force is applied to it over a distance. Here, the pitcher throws a baseball at a speed of \(40 \text{ m/s}\) and the catcher stops it with her glove. We need to find out if the work done by the catcher is positive, negative, or zero.

Define Work Concept

Work done on an object can be expressed as \( W = F \times d \times \cos\theta \), where \( F \) is the force applied, \( d \) is the distance over which the force is applied, and \( \theta \) is the angle between the force and displacement vectors. If the force and displacement are in opposite directions, the work is negative.

Analyze the Forces

The catcher applies a force to stop the baseball moving towards her; thus, the force is exerted in the opposite direction to the ball's motion. Hence, the angle \( \theta \) between the force applied by the catcher and the displacement of the ball is \(180^{\circ}\).

Determine the Work Done

Since the force applied by the catcher is opposite the direction of the ball's motion, \( \cos(180^{\circ}) = -1 \). Therefore, \( W = F \times d \times (-1) \), which results in a negative value for work. This indicates that the work done by the catcher on the baseball is negative.

Key concepts

Force

In physics, force is a fundamental concept that refers to an interaction that can change the motion of an object. In simpler terms, it's what moves or stops an object. When a pitcher throws a baseball, he applies a force to it with his throw.

The force is responsible for accelerating the baseball to a high speed of 40 m/s. However, once the baseball reaches the catcher, another force comes into play. The catcher applies a force in the opposite direction of the ball's motion to stop it. This force exerted by the catcher is crucial as it is responsible for stopping the baseball.

• Force is measured in newtons (N).
• It has both magnitude (how strong it is) and direction (which way it acts).
• When forces act on an object at rest, they can start its motion.

Understanding these aspects of force helps in analyzing experiences like catching a fast-moving baseball.

Displacement

Displacement is a term used to describe how far and in what direction an object has moved. It is different from distance as it considers the change in position of the object, not just the path covered.

In our scenario, the displacement of the baseball occurs when it travels from the pitcher's hand to the catcher's glove. During its motion, the baseball covers a certain distance, but displacement specifically refers to its initial and final locations.

• Measured in meters (m).
• Has both magnitude and direction, making it a vector quantity.
• It is important in calculating the work done since work depends on displacement.

Catching a baseball requires understanding its complete displacement, ensuring the force applied effectively stops the ball.

Angle between Force and Displacement

The angle between the force and displacement vectors is integral to understanding the work done on an object. It dictates how effective the force is in performing work. When the angle, denoted as \( \theta \), is 0 degrees, the force is in the same direction as the displacement, leading to maximum work. However, when \( \theta \) is 180 degrees, the force acts in the opposite direction to the displacement, affecting the work done.

In the case of the catcher and the baseball, the angle \( \theta \) is 180 degrees. This is because the catcher's force acts opposite to the direction of the baseball's displacement, which is moving forward.

• The formula for work includes \( \cos\theta \).
• Different angles lead to distinct work values:
  (0 degrees = positive work, 90 degrees = zero work, 180 degrees = negative work).

Understanding this angle and its effects is essential for comprehending concepts around motion and forces.

Negative Work

Negative work occurs when the direction of the force applied is opposite to the direction of displacement. In physics, work is calculated by the formula \( W = F \times d \times \cos\theta \). If the angle \( \theta \) is 180 degrees, and \( \cos(180^\circ) = -1 \), the work calculated will be negative.

With negative work, instead of adding energy to the system, energy is being removed. This is exactly what happens when the catcher stops the baseball—it removes the kinetic energy the ball had.

• Negative work does not mean bad; it simply means energy is taken away.
• The catcher’s negative work slows the ball, eventually bringing it to a stop.
• Energy conservation tells us the ball's kinetic energy decreases as the catcher's glove absorbs it.

Understanding negative work helps explain how forces can stop objects, showing an important application of the work-energy principle.