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Physics LibreTexts

4.9 Potential and Kinetic Energy

In astronomy, we encounter many examples of kinetic energy and gravitational potential energy. Every object moving in space has kinetic energy and everything that is subject to the gravity force of a star or planet has potential energy.

Examples of kinetic energy are all around us. We might imagine that a fast moving object has more kinetic energy that a similar object moving slowly. In fact, the kinetic energy is proportional to the square of the velocity. If we double the speed of a moving object, we increase the kinetic energy by a factor of four. We might also imagine that heavy or massive objects have more kinetic energy than light objects. In fact, kinetic energy is proportional to mass. These two relationships get combined in the simple equation for kinetic energy

Kinetic Energy = ½ mv2

In this equation, m is mass and v is velocity. If we insert velocity in meters per second and mass in kilograms, the kinetic energy comes out in units of Joules.

Let's look at a couple of examples of kinetic energy. A soccer ball weighs about 1 kilogram and can be kicked at about 15 meters per second. The kinetic energy is therefore ½ × 1 × (15 × 15) = 112.5 Joules. A baseball weighs about 250 grams or 0.25 kilograms and can be hurled at about 50 meters per second or 100 mph. A fastball has a kinetic energy of ½ × 0.25 × (50 × 50) = 312.5 Joules. Because of its speed, the small object has much more kinetic energy; you would be better off being hit by a soccer free kick than a fastball.

How about a human compared to a speeding bullet? A fast sprinter weighing 70 kilograms can run 100 meters in 10 seconds or a speed of 10 meters per second. This is a kinetic energy of ½ × 70 × (10 × 10) = 3500 Joules. A bullet has a mass of about 50 grams or 0.05 kilograms and travels with a typical velocity of about 400 meters per second or 900 mph. The bullet has a kinetic energy of ½ × 0.05 × (400 × 400) = 4000 Joules. A human really is nearly as "powerful" as a speeding bullet!

The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. When they start rising, the kinetic energy begins to be converted to gravitational potential energy. The sum of kinetic and potential energy in the system remains constant, ignoring losses to friction. Click here for original source URL.

Any object that is subject to the force of gravity will have an amount of gravitational potential energy. Here are some examples of gravitational potential energy: a barbell held over your head, a boulder balancing precariously on top of a hill, a roller coaster at the highest point in its path, a lake held above a valley by a dam. In each case something with mass has the potential to move under gravity. In each case the motion will be towards the source of that gravity, which is effectively the center of the Earth.

The gravitational potential energy is given by the weight or force of gravity exerted by an object multiplied by its height above the ground. Newton’s second law of motion says that gravity force is equal to mass times acceleration. Combining these relationships we see that

Gravitational Potential Energy = m g h

In this equation, m is mass, g is the Earth’s gravitational acceleration, and h is the height above the ground. We have already seen that acceleration at the Earth’s surface, denoted g, is 9.8 meters per second per second. A top weightlifter can lift 300 kilograms over his head. The potential energy of this weight lifted 2 meters off the ground is 300 × 9.8 × 2 = 5880 Joules. This is the amount of energy a human must provide to lift such a weight. When the weight is released, the gravitational potential energy is converted into kinetic energy as the weight falls.

Painting of?James Watt, the noted Scottish inventor and mechanical engineer by?Carl Frederik von Breda. Click here for original source URL.

Running or swinging a bat quickly takes more power than doing those tasks slowly. Scientists define power as the rate at which energy is expended. Mathematically, power is the energy generated divided by the time it takes to generate it. If the energy is expressed in units of Joules, the power is in Joules per second, or Watts. It is very appropriate that the unit of power is named after the James Watt, the Scottish inventor of the steam engine. This invention was the force behind the Industrial Revolution.

Consider the example of weightlifting. It takes only 2 seconds to raise the weight from the hanging position to the arms-raised position, a vertical distance of 1.5 meters. So in that second (1.5 / 2) × 5880 = 4410 Joules is generated, or a power of 4410 / 2 = 2205 Watts. This is enough to light up 20 light bulbs!

Here is a practical example of gravitational potential energy. Water that falls in the mountains can be harnessed to create energy. You could for example use a waterfall to turn a turbine and generate electricity. A modest sized waterfall might have a drop of 50 meters and a flow rate of 1000 gallons per second (this converts to about 4 meters cubed of water per second or 4000 kilograms per second). The available energy is therefore 4000 × 9.8 × 50 = 1.96 × 106 Joules each second or 1.96 million Watts, which is enough to power a small town.