1. Definition

The Work–Energy Theorem states:

The net work done on an object equals the change in its kinetic energy.

Mathematically:

\(W_{net}=\Delta K\)

where:

• \(W_{net}\)​ = net work done on the object
• \(Delta K\) = change in kinetic energy

2. Mathematical Form

Since kinetic energy is:

\(K=\frac{1}{2}mv^2\)

the theorem can be written as:

\(W_{net}=K_f-K_i\)​

or

\(W_{net}=\frac{1}{2}mv^2_f-\frac{1}{2}mv^2_i\)​

3. Physical Meaning

The theorem explains how forces change an object’s motion.

• Positive net work increases kinetic energy.
• Negative net work decreases kinetic energy.
• Zero net work leaves kinetic energy unchanged.

4. Positive Work

If the net force acts in the direction of motion:

\(W_{net}>0\)

then:

\(K_f>K_i\)​

The object speeds up.

Example:

• a car accelerating forward
• a falling object

5. Negative Work

If the net force opposes the motion:

\(W_{net}<0\)

then:

\(K_f<K_i\)​

The object slows down.

Example:

• friction on a sliding box
• vehicle braking

6. Zero Net Work

If:

\(W_{net}=0\)

then:

\(K_f=K_i\)​​

The object’s speed remains constant.

Example:

• motion at constant velocity on a frictionless surface

7. Derivation from Newton’s Second Law

Start with:

\(\sum F=ma\)

Using:

\(a=\frac{dv}{dt}\)​

and

\(v=\frac{dx}{dt}\)​​

we obtain:

\(a=v\frac{dv}{dx}\)​​

Substitute into Newton’s Second Law:

\(F=mv\frac{dv}{dx}\)​​

Multiply by $dx$:

\(F dx=mv dv\)​

Integrate both sides:

\(\int F dx=\int mv dv\)

The left side is work:

\(W_{net}\)​

The right side becomes:

\(\frac{1}{2}mv^2_f-\frac{1}{2}mv^2_i\)​

Therefore:

\(W_{net}=\Delta K\)

8. Relationship to Force–Position Graphs

For variable forces:

\(W=\int F(x) dx\)

The work equals the area under the force-versus-position graph.

Once work is found, the Work–Energy Theorem can be used to determine changes in speed.

9. Advantages of the Energy Method

Using the Work–Energy Theorem often avoids solving for:

• acceleration
• time
• individual components of motion

This makes many mechanics problems much simpler.

10. Example

A 2.0 kg object starts from rest.

A net force does 40 (J) of work.

Using:

\(W_{net}=\Delta K\)

\(40=\frac{1}{2}(2)v^2-0\)

\(40=v^2\)

\(v=\sqrt{40}\)

\(v\approx6.3m/s\)

11. Connection to Energy Conservation

The Work–Energy Theorem is always true.

Energy conservation is a special application that becomes useful when forces are conservative.

The theorem serves as the foundation for later topics such as:

• gravitational potential energy
• spring potential energy
• conservation of mechanical energy

Summary

The Work–Energy Theorem states:

\(W_{net}=\Delta K\)

or

\(W_{net}=\frac{1}{2}mv^2_f-\frac{1}{2}mv^2_i\)​​

Key ideas:

• positive work increases kinetic energy
• negative work decreases kinetic energy
• zero net work keeps kinetic energy constant
• derived directly from Newton’s Second Law

The Work–Energy Theorem provides one of the most powerful problem-solving tools in AP Physics C Mechanics because it directly connects force, work, and motion.