Work & Kinetic Energy

Push something along its path and you change its energy of motion by exactly the work done.

12 minUpdated 21 October 2026kinetic energyworkwork–energy theoremdot product

Lesson at a glanceShow the key points

Kinetic energy K = ½mv²; for a system, add the kinetic energies of all particles.
Work done by a force in a small displacement: dW = F·dr = F dr cos θ; over a path, W = ∫F·dr.
Only the component of force along the motion changes the speed; a force perpendicular to velocity does no work.
Work–energy theorem: the work done by the resultant force (= sum of works of all forces) equals Kf − Ki.

A sprinter at full speed carries something a sleeping cat does not. Physics calls it kinetic energy, and the way you give it to something is by doing work on it.

Kinetic energy

Definition · Kinetic energy

The energy a particle has because it moves, over and above its energy at rest:

K = \tfrac12 mv^2 = \tfrac12 m\,\vec v\cdot\vec v

. For a system, add up the kinetic energies of all its particles:

K = \sum \tfrac12 m_iv_i^2

Kinetic energy can grow, shrink or stay the same. With no force, velocity and hence K stay constant. And since K depends only on speed, a force can only change K if it has a component along the motion. A falling stone speeds up (gravity points along its velocity); a stone thrown upward slows down (gravity points against it); a ball whirled on a string keeps its speed (the tension is perpendicular to the velocity).

Make this precise. Differentiate K:

\frac{dK}{dt} = mv\frac{dv}{dt} = F_t\, v = F v\cos\theta = \vec F\cdot\vec v \quad\Rightarrow\quad dK = \vec F\cdot d\vec r \qquad (8.2)

where $F_t$ is the tangential component of the resultant force and θ is the angle between the force and the velocity.

Work

The quantity $\vec F\cdot d\vec r = F\,dr\cos\theta$ is the work done by the force during the small displacement $d\vec r$ . Over a finite path, add up the pieces:

Work done by a force(8.3)

W = \int \vec F\cdot d\vec r = \int F\cos\theta\,dr

integrated along the actual path; SI unit the joule (1 J = 1 N·m)

Interactive

Only the part of the force along the motion does work

A sledge moves 3 m to the right. Rotate the pull: pulling ahead does positive work, sideways does none, and pulling backwards does negative work.

F\cos\theta

16.38N

W = Fd\cos\theta

49.15J

Force

F

20N

Angle to the motion

\theta

35°

The work–energy theorem

If $\vec F$ is the resultant force, integrating (8.2) gives

Work–energy theorem

W_{\text{total}} = \int \vec F_{\text{net}}\cdot d\vec r = K_f - K_i

If several forces act, $\vec F = \vec F_1 + \vec F_2 + \dots$ , and the total work is the sum of the separate works $W_1 + W_2 + \dots$ . So the theorem says: the works done by all the forces add up to the change in kinetic energy.

Interactive

Every force's work adds up to the change in kinetic energy

A 5 kg crate starts at rest and is pushed horizontally for a distance d over a rough floor. Weight and the normal force are perpendicular to the motion, so they do no work.

120.0

push

-39.2

friction

0.0

gravity

0.0

normal

80.8

total = ΔK

\Delta K

80.80J

final speed5.69m/s

Push

F

30N

Friction coefficient

\mu_k

0.20

Distance

d

4.0m

Worked example

A dropped ball

A 1.5 kg ball is dropped from rest and falls 4.0 m. Ignoring air resistance, use the work–energy theorem to find its speed. (g = 9.8 m/s²)

What to remember

1 $K = \tfrac12 mv^2$ ; it changes only when a force has a component along the motion.
2Work: $W = \int \vec F\cdot d\vec r$ ; zero for a force perpendicular to the motion.
3Work–energy theorem: total work by all forces = Kf − Ki.

Check yourself

A car's speed doubles. Its kinetic energy becomes…

A satellite in a circular orbit: how much work does gravity do on it per orbit?

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

WatchIntroduction to work and energyKhan Academy WatchWork, Energy, and PowerCrash Course Physics #9 ReadWorkOpenStax University Physics §7.1 ReadKinetic energyOpenStax University Physics §7.2 ReadWork–energy theoremOpenStax University Physics §7.3 ReadDefinition and mathematics of workThe Physics Classroom ReadWorkHyperPhysics

Next Calculating Work

Work & Kinetic Energy

Push something along its path and you change its energy of motion by exactly the work done.

12 minUpdated 21 October 2026kinetic energyworkwork–energy theoremdot product

Lesson at a glanceShow the key points

Kinetic energy K = ½mv²; for a system, add the kinetic energies of all particles.
Work done by a force in a small displacement: dW = F·dr = F dr cos θ; over a path, W = ∫F·dr.
Only the component of force along the motion changes the speed; a force perpendicular to velocity does no work.
Work–energy theorem: the work done by the resultant force (= sum of works of all forces) equals Kf − Ki.

A sprinter at full speed carries something a sleeping cat does not. Physics calls it kinetic energy, and the way you give it to something is by doing work on it.

Kinetic energy

Definition · Kinetic energy

The energy a particle has because it moves, over and above its energy at rest:

K = \tfrac12 mv^2 = \tfrac12 m\,\vec v\cdot\vec v

. For a system, add up the kinetic energies of all its particles:

K = \sum \tfrac12 m_iv_i^2

Make this precise. Differentiate K:

\frac{dK}{dt} = mv\frac{dv}{dt} = F_t\, v = F v\cos\theta = \vec F\cdot\vec v \quad\Rightarrow\quad dK = \vec F\cdot d\vec r \qquad (8.2)

where $F_t$ is the tangential component of the resultant force and θ is the angle between the force and the velocity.

Work

The quantity $\vec F\cdot d\vec r = F\,dr\cos\theta$ is the work done by the force during the small displacement $d\vec r$ . Over a finite path, add up the pieces:

Work done by a force(8.3)

W = \int \vec F\cdot d\vec r = \int F\cos\theta\,dr

integrated along the actual path; SI unit the joule (1 J = 1 N·m)

Interactive

Only the part of the force along the motion does work

A sledge moves 3 m to the right. Rotate the pull: pulling ahead does positive work, sideways does none, and pulling backwards does negative work.

F\cos\theta

16.38N

W = Fd\cos\theta

49.15J

Force

F

20N

Angle to the motion

\theta

35°

The work–energy theorem

If $\vec F$ is the resultant force, integrating (8.2) gives

Work–energy theorem

W_{\text{total}} = \int \vec F_{\text{net}}\cdot d\vec r = K_f - K_i

Interactive

Every force's work adds up to the change in kinetic energy

A 5 kg crate starts at rest and is pushed horizontally for a distance d over a rough floor. Weight and the normal force are perpendicular to the motion, so they do no work.

120.0

push

-39.2

friction

0.0

gravity

0.0

normal

80.8

total = ΔK

\Delta K

80.80J

final speed5.69m/s

Push

F

30N

Friction coefficient

\mu_k

0.20

Distance

d

4.0m

Worked example

A dropped ball

A 1.5 kg ball is dropped from rest and falls 4.0 m. Ignoring air resistance, use the work–energy theorem to find its speed. (g = 9.8 m/s²)

What to remember

1 $K = \tfrac12 mv^2$ ; it changes only when a force has a component along the motion.
2Work: $W = \int \vec F\cdot d\vec r$ ; zero for a force perpendicular to the motion.
3Work–energy theorem: total work by all forces = Kf − Ki.

Check yourself

A car's speed doubles. Its kinetic energy becomes…

A satellite in a circular orbit: how much work does gravity do on it per orbit?

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

Next Calculating Work

Kinetic energy#

Work#

The work–energy theorem#

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