Problem 26 - University Physics with Modern Physics

Question

A batter hits a baseball with mass 0.145 \(\mathrm{kg}\) straight upward with an initial speed of 25.0 \(\mathrm{m} / \mathrm{s}\) . (a) How much work has gravity done on the baseball when it reaches a height of 20.0 \(\mathrm{m}\) above the bat? (b) Use the work-energy theorem to calculate the speed of the baseball at a height of 20.0 m above the bat. You can ignore air resistance. (c) Does the answer to part (b) depend on whether the baseball is moving upward or downward at a height of 20.0 \(\mathrm{m} ?\) Explain.

Step-by-Step Solution

Answer

(a) Work done by gravity: 28.449 J. (b) Speed at 20m: 17.92 m/s. (c) The speed is the same regardless of direction at that height.

Step 1: Identify given values and formulas

We are given the mass of the baseball, \( m = 0.145 \ \text{kg} \), the initial speed, \( v_i = 25.0 \ \text{m/s} \), and the height \( h = 20.0 \ \text{m} \). The acceleration due to gravity \( g = 9.81 \ \text{m/s}^2 \). We will apply the work done by gravity formula and work-energy theorem.

Step 2: Calculate the work done by gravity

The work done by gravity is given by \( W = mgh \), where \( m \) is the mass, \( g \) is acceleration due to gravity, and \( h \) is the height. Substitute the values:\[ W = (0.145 \ \text{kg})(9.81 \ \text{m/s}^2)(20.0 \ \text{m}) \]Calculate \( W \) to find the work done by gravity.

Step 3: Apply the work-energy theorem

The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy: \( W = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 \). From Step 2, we have \( W = mgh \), thus:\[ mgh = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 \]Rearranging the equation to solve for \( v_f \), the final speed at height \( h \).

Step 4: Solve for the final speed

First, express \( v_f \) from the equation obtained:\[ \frac{1}{2}mv_f^2 = mgh + \frac{1}{2}mv_i^2 \]\[ v_f^2 = \frac{2mgh}{m} + v_i^2 \]\[ v_f^2 = 2gh + v_i^2 \]\[ v_f = \sqrt{2gh + v_i^2} \]Substitute the given values:\[ v_f = \sqrt{2(9.81 \ \text{m/s}^2)(20.0 \ \text{m}) + (25.0 \ \text{m/s})^2} \]Calculate \( v_f \).

Step 5: Analyze direction's impact on speed

The work-energy theorem considers speed without direction. Thus, whether the object is moving upward or downward at a given height doesn't change the speed at that height, assuming no other forces are acting, such as air resistance.

Key Concepts

Gravity

Gravity is a fundamental force in physics, playing a big role in the work-energy theorem. It is the force that pulls objects towards the Earth.
In this problem, we consider gravity's impact on a baseball hit upwards. Gravity does negative work as it acts in the opposite direction to the baseball's initial motion.
When calculating the work done by gravity, we use the formula

• \( W = mgh \) where \( m \) is mass, \( g \) is the acceleration due to gravity (\( 9.81 \, \text{m/s}^2 \)), and \( h \) is the height.

This work done by gravity reduces the kinetic energy of the baseball as it rises, converting it into potential energy. Understanding gravity helps us to track energy changes in motion and solve problems involving heights and speeds.

Kinetic Energy

Kinetic energy refers to the energy an object has due to its motion. It's a core part of the work-energy theorem.
The formula for kinetic energy is:

• \( KE = \frac{1}{2}mv^2 \)

where \( m \) is the mass and \( v \) is the velocity of the object.
In the baseball problem, as it is hit upwards, it starts with kinetic energy from its initial speed.
When the baseball reaches a certain height, the kinetic energy changes, influenced by the work done by gravity.
This change in kinetic energy helps us find the new speed of the baseball using the work-energy theorem. By understanding kinetic energy, you can comprehend how objects move and how their speed varies with energy input or output.

Potential Energy

Potential energy is often associated with an object's position relative to Earth.
In the context of gravity:

• Potential energy due to height is given by \( PE = mgh \).

As the baseball rises to a higher altitude, its potential energy increases while the kinetic energy decreases.
The work done against gravity is stored as potential energy.
In our baseball example, the calculation of work done by gravity helps us to determine the change in potential energy.
This is especially vital when applying the work-energy theorem.

• Potential energy is crucial in understanding how objects can store energy based on their position.

It serves as essential background to analyze motion and energy dynamics in mechanics.

Mechanics

Mechanics is the branch of physics dealing with the motion and forces acting on objects.
Understanding mechanics is crucial for applying the work-energy theorem effectively. In our baseball scenario, the interplay between forces like gravity and energy forms the core analytical basis.
Here’s what we consider in mechanics while solving this problem:

• The force of gravity acting on the baseball, influencing its speed and energy distribution.
• Application of the work-energy theorem to determine the object's speed at different points in its path.
• Realizing that mechanics doesn't account for directionality of speed in energy calculations, as seen with the baseball moving upwards or downwards.

Thus, mechanics teaches us to view motion in terms of energy shifts between kinetic and potential forms, particularly in gravitational contexts.