# How is Height Related to Potential Energy?

Potential energy is the energy stored in an object due to its position, state, or condition. It is the energy that an object has the potential to use or release as work. One of the most common types of potential energy is gravitational potential energy, which depends on the height and mass of an object.

## What is Gravitational Potential Energy?

Gravitational potential energy is the energy that an object has because of its position in a gravitational field. The higher an object is, the more gravitational potential energy it has. This is because the object has more potential to fall and do work on another object or system.

The formula for gravitational potential energy is:

��=��ℎPE=mgh

Where:

• PE is the potential energy in joules (J)
• m is the mass of the object in kilograms (kg)
• g is the acceleration due to gravity, which is about 9.8 meters per second squared (m/s^2) on Earth
• h is the height of the object above a reference point in meters (m)

For example, a 1 kg ball that is 10 m above the ground has a gravitational potential energy of:

��=1×9.8×10PE=1×9.8×10

��=98�PE=98J

## How Does Height Affect Potential Energy?

According to the formula, the gravitational potential energy of an object is directly proportional to its height. This means that if the height of an object increases, its potential energy also increases. Conversely, if the height of an object decreases, its potential energy also decreases.

For example, if we double the height of the ball in the previous example, its potential energy will also double:

��=1×9.8×20PE=1×9.8×20

��=196�PE=196J

Similarly, if we halve the height of the ball, its potential energy will also halve:

��=1×9.8×5PE=1×9.8×5

��=49�PE=49J

This shows that height and potential energy have a linear relationship. A graph of height versus potential energy would look like a straight line with a positive slope.

## How Does Potential Energy Convert to Kinetic Energy?

When an object falls from a certain height, it loses some of its potential energy and gains some kinetic energy. Kinetic energy is the energy that an object has because of its motion. The faster an object moves, the more kinetic energy it has.

The formula for kinetic energy is:

��=12��2KE=21​mv2

Where:

• KE is the kinetic energy in joules (J)
• m is the mass of the object in kilograms (kg)
• v is the velocity or speed of the object in meters per second (m/s)

For example, if we drop the ball from a height of 10 m, it will start with a potential energy of 98 J and zero kinetic energy. As it falls, it will lose some of its potential energy and gain some kinetic energy. When it reaches the ground, it will have zero potential energy and a maximum kinetic energy.

According to the law of conservation of energy, the total energy of a system remains constant. This means that the potential energy that the ball loses must be equal to the kinetic energy that it gains. Therefore, we can write:

���=���PEi​=KEf​

Where:

• PE_i is the initial potential energy
• KE_f is the final kinetic energy

Substituting the values for PE_i and KE_f, we get:

98=12×1×�298=21​×1×v2

Solving for v, we get:

�=1961v=1196​​

�=14�/�v=14m/s

This means that when the ball reaches the ground, it will have a speed of 14 m/s.

## Conclusion

Height and potential energy are related by a direct proportionality. The higher an object is, the more gravitational potential energy it has. When an object falls from a certain height, it converts some of its potential energy into kinetic energy. The total energy of a system remains constant according to the law of conservation of energy.