Work-Energy Theorem Formula

The Formula

W_{\text{net}} = \Delta KE = KE_{\text{final}} - KE_{\text{initial}}

When to use: The total work done on an object is exactly what changes its kinetic energy.

Quick Example

Push a cart (do positive work) → it speeds up (gains KE). Friction (negative work) → it slows down (loses KE).

Notation

W_{\text{net}} is the net work in joules (J), \Delta KE is the change in kinetic energy, m is mass in kg, v_i and v_f are initial and final speeds in m/s.

What This Formula Means

The net work done on an object by all forces acting on it equals the change in its kinetic energy.

The total work done on an object is exactly what changes its kinetic energy.

Formal View

The work-energy theorem states W_{\text{net}} = \int \vec{F}_{\text{net}} \cdot d\vec{s} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2. It is derived directly from Newton's second law by integrating \vec{F} = m\vec{a} along the displacement.

Worked Examples

Example 1

easy
A 4 \text{ kg} box initially at rest is pushed with a net force of 20 \text{ N} over 5 \text{ m}. What is the final speed of the box?

Solution

  1. 1
    The work-energy theorem states: W_{\text{net}} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2.
  2. 2
    Net work done: W = Fd = 20 \times 5 = 100 \text{ J}.
  3. 3
    Since v_i = 0: 100 = \frac{1}{2}(4)v_f^2 \implies v_f = \sqrt{\frac{200}{4}} = \sqrt{50} \approx 7.07 \text{ m/s}

Answer

v_f \approx 7.07 \text{ m/s}
The work-energy theorem directly connects the net work done on an object to its change in kinetic energy. It provides an alternative to using kinematics equations for finding final speeds.

Example 2

medium
A 1500 \text{ kg} car traveling at 25 \text{ m/s} brakes with a friction force of 7500 \text{ N}. How far does it take to stop?

Common Mistakes

  • Using the work done by only one force instead of the net work — W_{\text{net}} must include work from all forces (applied, friction, gravity, etc.).
  • Forgetting that negative work reduces kinetic energy — friction does negative work, so it decreases the object's KE.
  • Confusing the work-energy theorem with conservation of energy — the theorem relates net work to KE change, while conservation of energy includes all energy forms (PE, thermal, etc.).

Why This Formula Matters

The work-energy theorem lets you solve problems where forces vary with position (like springs) or where tracking acceleration is complicated. It is often easier to calculate work done than to integrate forces over time.

Frequently Asked Questions

What is the Work-Energy Theorem formula?

The net work done on an object by all forces acting on it equals the change in its kinetic energy.

How do you use the Work-Energy Theorem formula?

The total work done on an object is exactly what changes its kinetic energy.

What do the symbols mean in the Work-Energy Theorem formula?

W_{\text{net}} is the net work in joules (J), \Delta KE is the change in kinetic energy, m is mass in kg, v_i and v_f are initial and final speeds in m/s.

Why is the Work-Energy Theorem formula important in Physics?

The work-energy theorem lets you solve problems where forces vary with position (like springs) or where tracking acceleration is complicated. It is often easier to calculate work done than to integrate forces over time.

What do students get wrong about Work-Energy Theorem?

Only net work changes KE—individual forces may do positive or negative work.

What should I learn before the Work-Energy Theorem formula?

Before studying the Work-Energy Theorem formula, you should understand: work, kinetic energy.

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Related Concepts

WorkKinetic EnergyPower