Work-Energy Theorem

Energy
principle

Also known as: work-kinetic energy theorem

Grade 9-12

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The net work done on an object by all forces acting on it equals the change in its kinetic energy. The work-energy theorem lets you solve problems where forces vary with position (like springs) or where tracking acceleration is complicated.

Definition

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

💡 Intuition

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

🎯 Core Idea

This theorem bridges force-based thinking (work) with energy-based thinking (kinetic energy).

Example

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

Formula

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

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.

🌟 Why It 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.

💭 Hint When Stuck

When using the work-energy theorem, first identify all forces doing work on the object. Then calculate the net work: add positive work (forces in the direction of motion) and subtract negative work (forces opposing motion). Finally, set W_{\text{net}} = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 and solve for the unknown.

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.

Related Concepts

WorkKinetic EnergyPower

🚧 Common Stuck Point

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

⚠️ 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.).

Frequently Asked Questions

What is Work-Energy Theorem in Physics?

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

What is the Work-Energy Theorem formula?

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

When do you use Work-Energy Theorem?

When using the work-energy theorem, first identify all forces doing work on the object. Then calculate the net work: add positive work (forces in the direction of motion) and subtract negative work (forces opposing motion). Finally, set W_{\text{net}} = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 and solve for the unknown.

Prerequisites

workkinetic energy

Next Steps

power

How Work-Energy Theorem Connects to Other Ideas

To understand work-energy theorem, you should first be comfortable with work and kinetic energy. Once you have a solid grasp of work-energy theorem, you can move on to power.

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