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Explanation of the Relationship between Work and Kinetic Energy by Ron Kurtus - Succeed in Understanding Physics. Also refer to physical science, force, distance, mass acceleration, velocity, equation, newton, kilogram, joule, Ron Kurtus, School for Champions. Copyright © Restrictions
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.
Physical Science rules
Resources and references
The following resources provide information on this subject:
Websites
Books
Top-rated
books on the Science of Work
Mini-quiz to check your understanding
If you got all three correct, you are on your way to becoming a Champion in Physics. If you had problems, you had better look over the material again.
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Relationship between Work and Kinetic Energy