Question

Triple Choice A pitcher throws a baseball at 40 m>s (~90 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 on the ball by the catcher is negative.

Step-by-step solution

Understanding Work on an Object

Work done on an object is given by the formula: \[Work = Force \times Displacement \times \cos(\theta)\]where \(\theta\) is the angle between the force and the displacement vector. If the force is in the opposite direction to the displacement, \(\cos(\theta) = -1\), resulting in negative work.

Analyzing the Catching Process

When the pitcher throws the ball, it moves towards the catcher's glove. As the ball is caught, the glove applies a force in the opposite direction of the ball's motion to stop it. This force opposes the ball's displacement at each point until it comes to rest.

Calculating the Work Done

Since the force exerted by the catcher on the ball is opposite to the direction of the ball's velocity and displacement (as the ball moves towards the catcher and stops), the angle \(\theta\) between the force and the displacement is 180 degrees, leading to \(\cos(180^\circ) = -1\). Therefore, the work done on the ball is negative.

Key concepts

Force

Force is fundamentally a push or pull action that acts upon an object as a result of its interaction with another object. We encounter various types of forces daily, including gravitational, frictional, and applied forces. In the scenario of catching a baseball, the glove applies an opposing force to the moving ball to bring it to a halt.
Understanding how force works in such situations is crucial. The magnitude of force affects how quickly an object can change its speed or direction.

• The force exerted by the catcher is intentionally applied against the ball's velocity.
• This opposing force is what stops the ball.

In classroom examples, forces are often portrayed with vectors. These vectors have both magnitude and direction, helping us visualize the interactions between forces and objects.
Mastering the vector representation is important, as it allows us to understand the resulting motion from a given set of forces acting upon an object.

Displacement

Displacement refers to the change in position of an object. It's a vector quantity, meaning it has both magnitude and direction, similar to force. In the activity of catching a baseball, we can see that displacement is vital to understanding how the ball moves from the pitcher's hand to the catcher's glove.
To analyze the catch:

• The ball travels a certain distance in a specific direction from the pitcher's hand to the catcher.
• We call this directed distance 'displacement'.

This characteristic of displacement, distinguishing it from mere distance, incorporates direction, setting it apart in its applications.
During the act of catching, the displacement of the ball becomes progressively zero once the ball stops moving altogether. Thus, the catcher’s glove is involved in reducing the ball's displacement by exerting a force in the opposite direction.

Negative Work

Negative work occurs when the force applied to an object is in the direction opposite to its displacement. In our baseball scenario, as the catcher exerts a force in the opposite direction to the ball's movement, this interplay results in negative work.
Let's break this down:

• Since work done is given by the formula \[ Work = Force \times Displacement \times \cos(\theta) \] where \( \theta \) is the angle between the force and displacement vectors.
• If this angle is 180 degrees, as in this case, then the cosine of 180 degrees equals -1.

This tells us that the work done by the catcher's force on the ball is negative because the force and displacement are in precisely opposite directions.
Negative work indicates that energy is being extracted from the moving object, effectively reducing its motion until it comes to a stop. In practical scenarios, negative work is common in stopping mechanisms found in vehicles, sports, and various engineering applications.