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## Proof that Work Can be the Change in Kinetic Energy

by Ron Kurtus (6 October 2013)

When you accelerate an object, you are doing work against inertia. That work equals the change in kinetic energy of the object.

Proof that the relationship is true can be determined by starting with the standard equation for work against inertia. You then define the constant acceleration in terms of velocity and time, as well as the distance traveled.

By some simple algebraic substitution, you can derive the equation for work in terms of change in kinetic energy.

Questions you may have include:

- What do you want to prove?
- Where do you start?
- How are terms manipulated to get the result?

This lesson will answer those questions. Useful tool: Units Conversion

## Show work equals change in KE

You want to prove that the equation for work in terms of the change in kinetic energy of an object is:

W = ΔKE

or

W = KE_{f}− KE_{i}

where

**W**is the work done against the resistance of inertia**ΔKE**is the change in kinetic energy (**Δ**is Greek letter capital delta)**KE**is the final kinetic energy of the object (_{f}**KE**)_{f}= mv_{f}^{2}/2**KE**is the initial kinetic energy of the object (_{i}**KE**)_{i}= mv_{i}^{2}/2

## Start with standard work against inertia equation

When you accelerate an object, you are doing work against inertia over the distance that the object is accelerated:

W = mad

where

**W**is the work in joules (J or kg-m²/s²)**m**is the mass of the object in kg**a**is the acceleration of the object in m/s²**d**is the distance the object moves in meters (m)

## Break into components to prove equation

If the acceleration is constant, it is equal to the change in velocity over time:

a = (v_{f}− v_{i})/t

This can be rearranged to be:

t = (v_{f}− v_{i})/a

Also, the distance traveled is the product of the average of the velocities and time:

d = t(v_{f}+ v_{i})/2

Substitute **t** in the equation:

d = (v_{f}− v_{i})(v_{f}+ v_{i})/2a

Since **(v _{f} − v_{i})(v_{f} + v_{i}) = v^{2}_{f} − v^{2}_{i}**, you get:

2ad = v^{2}_{f}− v^{2}_{i}

Multiply both sides of the equation by **m** and divide by **2**:

mad = m(v^{2}_{f}− v^{2}_{i})/2

mad = mv^{2}_{f}/2 − mv^{2}_{i}/2

W = KE_{f}− KE_{i}

∴ W = ΔKE(

∴means "therefore")

## Summary

You can show that work equals the change in kinetic energy of the object by starting with the standard equation for work against inertia. You define the constant acceleration in terms of velocity and time, as well as the distance traveled. Then by simple algebraic substitution, you can prove the equation for work in terms of change in kinetic energy.

Be clever

## Resources and references

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**Top-rated books on Physics of Work**

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## Proof that Work Can be the Change in Kinetic Energy