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Relationship between Work and Kinetic Energy

by Ron Kurtus (revised 13 October 2007)

There is a relationship between work and kinetic energy. Work is defined in Physical Science as the result of a force moving an object a distance and is stated by the equation W = Fd. But the result of the force being applied on the object also means that the object is moving with some given velocity, according to the equation for force as F = ma. From those two equations, it can be shown that work is equivalent to kinetic energy KE = ½mv².

Questions you may have include:

• What does W = Fd mean?
• What does F = ma mean?
• How does work equal kinetic energy?

This lesson will answer those questions. There is a mini-quiz near the end of the lesson.

Useful tools: Metric-English Conversion | Scientific Calculator.

Work is force times distance

The definition of work is that it equals force times the distance traveled while that force is being applied or

W = Fd

where:

• W is the work in joules (J or kg-m²/s²)
• F is the force applied to an object in newtons (N or kg-m/s²)
• d is the distance the object moves in meters (m)
• Fd is F times d

Thus, if you would apply a force of F = 3 newtons (3 N) to move an object d = 5 meters (5 m), the work done would be W = Fd = 15 joules (15 J).

The force required to do some work is determined from the equation F = W/d.

Force is mass times acceleration

We know that a force will accelerate an object according to the equation

F = ma

where:

• F is the force in newtons (N)
• m is the mass of the object in kilograms (kg)
• a is the acceleration or change in velocity in meters per second-squared (m/s²)
• ma is m times a

Rewriting the equation, the acceleration is a = F/m. Thus, if F = 3 N and m = 2 kg, then the acceleration would be a = 1.5 m/s².

Work is kinetic energy

Work is related to kinetic energy. This can be shown by examining the force and time it takes to travel a distance.

Acceleration

Acceleration is defined as the change in velocity over a time period. Suppose you start with an object at rest and accelerate it up to some end velocity v. The change in velocity is (v − 0) = v. Thus, the acceleration in such a case is

a = (v − 0)/t = v/t

where:

• v is the end velocity in meters per second
• t is the time to go from zero velocity to the end velocity in seconds
• v/t is v divided by t

If we substitute a = v/t into the equation F = ma, we get F = mv/t.

Distance

Since the velocity went from 0 to v, then v/2 is the average velocity for the motion.

The distance traveled equals the average velocity times the time, so we get d = ½vt for the distance traveled when the force is applied.

Kinetic energy

Now let's take all of the equations together.

Since W = Fd and F = mv/t, we get W = mvd/t.

And since d = ½vt, we substitute it into the equation and get W = mv(½vt)/t. Since t/t = 1, the equation becomes W = ½mv², which is the equation for kinetic energy. KE = ½mv². Pretty clever!

Summary

It can be shown that there is a relationship between work and kinetic energy. Work is the result of a force moving an object a distance and is stated by the equation W = Fd. The result of the force being applied on the object means that the object is moving with some given velocity, according to the equation F = ma. From those two equations, it can be shown that work is equivalent to kinetic energy KE = ½mv².

Answers to Readers' Questions

Physical Science rules

Resources

The following resources provide information on this subject:

Websites

Physical Science Resources

Books

Top-rated books on the Science of Work

Miscellaneous

Mini-quiz to check your understanding

1. If the work done to move a 2 kg object 1 m was 10 J, how much would the force be?

2. If an object started at 2 m/s and accelerated to 8 m/s in 3 seconds, what would the acceleration be?

3. If a force of 3 N moved a 6 kg object 4 m, what would its kinetic energy be?

If you got all three correct, you are on your way to becoming a Champion in Physical Science. If you had problems, you had better look over the material again.

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