1. What Is the Work-Energy Theorem?
The work-energy theorem states that:
The net work done on an object is equal to the change in its kinetic energy.
In symbols:
\( W_{net} = \Delta KE = KE_2 - KE_1 \)
This means that when forces do work on an object, they change how fast the object is moving.
1.1. Why This Theorem Is Useful
Instead of analysing forces one by one, we can directly relate work to changes in speed. This makes many physics problems easier to understand and solve.
2. Understanding the Theorem with Simple Logic
If you push an object and it speeds up, you have increased its kinetic energy. If you apply a force to slow something down, you reduce its kinetic energy. The work-energy theorem captures this idea in a simple equation.
2.1. When Work Is Positive
Positive work increases kinetic energy. The object speeds up.
- Pushing a cart forward
- A person accelerating while running
- A car speeding up when the accelerator is pressed
2.2. When Work Is Negative
Negative work decreases kinetic energy. The object slows down.
- Brakes on a car
- Friction slowing a sliding object
- A person trying to stop a moving ball
3. Deriving the Work-Energy Theorem (Simple Version)
The theorem comes from combining Newton’s second law and equations of motion.
3.1. Step-by-Step Derivation
Start with work done by a constant force:
\( W = Fd \)
But from Newton’s second law, \( F = ma \).
Use the equation of motion:
\( v^2 = u^2 + 2ad \)
Rearrange:
\( d = \dfrac{v^2 - u^2}{2a} \)
Substitute into \( W = Fd = ma d \):
\( W = ma \cdot \dfrac{v^2 - u^2}{2a} = \dfrac{1}{2}mv^2 - \dfrac{1}{2}mu^2 \)
So,
\( W = KE_2 - KE_1 \)
4. Examples That Show the Theorem
The work-energy theorem is easy to see in everyday motion.
4.1. Example 1: Pushing a Box
If you push a box and it starts moving faster, the positive work done by your push increases its kinetic energy.
4.2. Example 2: Applying Brakes
When brakes apply a force opposite to motion, negative work reduces the car’s kinetic energy and brings it to a stop.
4.3. Example 3: A Ball Rolling on the Floor
Friction does negative work, reducing the ball’s kinetic energy until it eventually stops.
5. Significance of the Work-Energy Theorem
This theorem is extremely useful because it connects force, motion and energy in a single expression. Instead of tracking individual forces, we can understand a system simply by looking at how energy changes.
5.1. Where It Is Commonly Used
- Vehicle acceleration and braking
- Objects falling under gravity
- Sports like cricket, tennis or football
- Analysing motion on inclined planes
- Solving many numerical problems in physics