What is the formula for work done by a force?

W = F · d · cos θ, where F is the applied force in newtons, d is the displacement in metres and θ is the angle between the force vector and the direction of motion. When the force and motion are parallel (θ = 0°) cos θ = 1 and the formula simplifies to W = F · d. The unit is the joule (J = N·m).

How is kinetic energy related to velocity?

KE = ½ m v². Because velocity is squared, doubling the speed quadruples the kinetic energy. A 1,000 kg car at 30 m/s has KE = 450,000 J (450 kJ); at 60 m/s it has 1,800 kJ — four times as much. This is why braking distances increase dramatically with speed.

What is the difference between gravitational PE (mgh) and elastic PE (½kx²)?

Gravitational potential energy (PE = mgh) is stored by lifting a mass m through height h against gravity g = 9.81 m/s². Elastic potential energy (PE = ½kx²) is stored when a spring with constant k is stretched or compressed by distance x from its natural length — a direct consequence of Hooke's law F = kx. Both are recoverable forms of energy that convert to kinetic energy when released.

What does the work-energy theorem say?

The net work done on an object equals its change in kinetic energy: W_net = ΔKE = ½mv_f² − ½mv_i². This is a powerful shortcut — you do not need to know the force explicitly if you know the initial and final speeds. Positive net work means the object accelerates; negative net work means it decelerates.

How is an elastic collision different from a perfectly inelastic one?

In an elastic collision both momentum and kinetic energy are conserved — the two objects bounce apart. The final velocities are v₁ = ((m₁−m₂)u₁ + 2m₂u₂)/(m₁+m₂) and v₂ = ((m₂−m₁)u₂ + 2m₁u₁)/(m₁+m₂). In a perfectly inelastic collision the objects stick together and move at a common final velocity v = (m₁u₁+m₂u₂)/(m₁+m₂); kinetic energy is not conserved — some is converted to heat, sound or deformation.

What is centripetal force and how is it calculated?

Centripetal force is the net inward force that keeps an object moving in a circle. F = mv²/r, where m is mass, v is the tangential speed and r is the radius. It is not a new type of force — it is simply the label for whatever force (tension, gravity, friction, normal) acts centripetally. The centripetal acceleration is a = v²/r, directed toward the centre.

Work & Energy Calculator

The Work & Energy Calculator is a ten-in-one physics tool that covers every major formula in classical mechanics relating to energy, force and motion. Whether you are a student revising for an A-level paper, an engineer checking a quick power budget, or a teacher preparing examples, the calculator shows the full algebraic working so every answer is self-explanatory and reproducible.

What the calculator covers

Each tab addresses a distinct physical concept, with solve-for-variable support so you can rearrange formulas without algebra:

Work done — W = F · d · cos θ. Solve for work, force, distance or the angle between force and displacement.
Kinetic energy — KE = ½mv². Solve for KE, mass or velocity; useful for checking speed limits and impact energies.
Gravitational PE — PE = mgh. Adjustable g lets you switch between Earth (9.81 m/s²), the Moon (1.62 m/s²) or any other body.
Elastic PE — PE = ½kx² (Hooke’s law energy). Solve for stored energy, spring constant or extension.
Power — P = W/t and P = F·v. Two methods — useful when you know either the time taken or the instantaneous velocity.
Momentum — p = mv. Solve for momentum, mass or velocity; foundation of Newton’s second law in integral form.
Impulse — J = F·Δt = Δp. Side-by-side panels show both the force-time and the momentum-change routes.
Collisions — elastic (both KE and momentum conserved) and perfectly inelastic (objects stick together). Shows all final velocities plus the KE lost.
Centripetal force — F = mv²/r. Also shows centripetal acceleration a = v²/r.
Work-energy theorem — W_net = ΔKE. Directly links net work to the change in kinetic energy.

How the physics works

Work and the angle factor

When a force is applied at an angle to the direction of motion, only its component along the displacement does useful work. The cosine factor captures this: a force at 90° does zero work (e.g. a book sitting on a table, where the normal force is perpendicular to motion). A negative angle (or angle greater than 90°) gives negative work — the force opposes motion.

Energy conservation

The principle of energy conservation underpins every tab. Gravitational PE converts to KE as an object falls (mgh = ½mv²), elastic PE converts to KE as a spring releases, and net work done on an object equals its kinetic energy gain. The collision tab makes the contrast vivid: elastic collisions preserve total KE; inelastic collisions do not — the deficit shows exactly how much energy was converted to internal energy (heat, deformation, sound).

Constants used

g = 9.81 m/s² (Earth surface, adjustable on the PE tab)
h = 6.626 × 10⁻³⁴ J·s (Planck constant — relevant for de Broglie wavelength; not in this tool)
All other constants (k, m, v, F, r) are entered by the user.

Worked example

A 1,200 kg car brakes from 25 m/s to rest:

Initial KE = ½ × 1,200 × 25² = 375,000 J
Work done by brakes = ΔKE = 0 − 375,000 = −375,000 J (negative: force opposes motion)
If the braking distance is 40 m, the average braking force = 375,000 ÷ 40 = 9,375 N

Use the Work (W=Fd) tab with W = 375,000, d = 40 and θ = 180° to verify: F = 375,000 ÷ (40 × cos 180°) = 9,375 N.

Formula reference

Quantity	Formula	Units
Work	W = F · d · cos θ	J
Kinetic energy	KE = ½ m v²	J
Gravitational PE	PE = m · g · h	J
Elastic PE	PE = ½ k x²	J
Power (work/time)	P = W ÷ t	W
Power (force × velocity)	P = F · v	W
Momentum	p = m · v	kg·m/s
Impulse	J = F · Δt = Δp	N·s
Centripetal force	F = m v² ÷ r	N
Work-energy theorem	W_net = ½mv_f² − ½mv_i²	J

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