Question

Analyze You throw a ball straight up into the air. It reaches a maximum height and returns to your hand. At what location(s) is the kinetic energy of the ball (a) a maximum and (b) a minimum? At what location(s) is the potential energy of the ball (c) a maximum and (d) a minimum?

Short answer

At maximum kinetic energy: just after release and just before catching; at minimum kinetic energy: maximum height. At maximum potential energy: maximum height; at minimum potential energy: at the initial and final positions.

Step-by-step solution

Understanding the Energy Principles

Kinetic energy (KE) is highest when the ball's speed is greatest, and potential energy (PE) is highest at the highest point of the ball's motion due to gravitational potential energy.

Identifying Maximum Kinetic Energy

The kinetic energy of the ball is maximum at two points: just as it is thrown and when it returns to the hand. These are the points where its velocity is greatest.

Identifying Minimum Kinetic Energy

The kinetic energy of the ball is minimum at the maximum height of its trajectory, where its velocity is zero for an instant.

Identifying Maximum Potential Energy

The potential energy of the ball is maximum at the highest point of its trajectory, as potential energy is related to height above the ground.

Identifying Minimum Potential Energy

The potential energy of the ball is minimum when the ball is at the lowest point in its path, i.e., at the same level as where it was initially thrown and caught.

Key concepts

Kinetic Energy

Kinetic energy is a fascinating concept that helps us understand how objects move. Imagine throwing a ball into the air. The kinetic energy of the ball is highest at the exact moments it's first launched and when it falls back into your hand. This is when it is moving the fastest.
When you give an object a push, like the ball, you're providing it with kinetic energy. This energy is directly related to how fast the object is moving and its mass. More speed or mass means more kinetic energy. We use the formula:

• \( KE = \frac{1}{2}mv^2 \)

where \( m \) is the mass and \( v \) is the velocity.
When the ball reaches the top of its flight and stops for a split second, its speed is zero, and so is its kinetic energy.

Potential Energy

Potential energy is energy stored within an object, ready to be transformed into kinetic energy or work. Think of it like a spring that's been compressed or a bow that has been drawn. In our example with the ball, when it is at the highest point of its trajectory, it possesses the maximum potential energy.
This energy is potential because the ball is capable of falling down, converting this stored energy into movement (kinetic energy) as it descends. As the ball rises, it slows down, exchanging kinetic energy for potential energy, but at the peak, it stops, and all the energy is potential.

• This potential energy depends on the height of the object and can be calculated with:\( PE = mgh \)

where \( m \) is mass, \( g \) is the gravitational acceleration (\( 9.81 \text{ m/s}^2 \) on Earth), and \( h \) is height above the ground.

Gravitational Potential Energy

Gravitational potential energy is a specific type of potential energy that is closely tied to the height of an object above the ground. Whenever you elevate an object against Earth's gravity, you give it gravitational potential energy. In the context of our ball, this energy peaks when the ball is at its highest point.
This is because gravitational potential energy is proportional to the object's height and its weight. Using the formula:

• \( PE = mgh \)

you can see that a higher \( h \) (height) means more potential energy.
When the ball reaches its maximum height, it stores the most gravitational potential energy. As it falls, this energy gets converted back into kinetic energy, making the ball move faster towards the ground. When the ball returns to the starting height, its gravitational potential energy is back to its minimum, ready to become kinetic energy again.