Work and energy

Why this matters

Forces and accelerations tell you what happens from moment to moment. Energy lets you skip the middle and jump straight from the start to the end. How fast is a roller coaster at the bottom of a drop? How much electricity does a pump need to lift water to a tank? How much heat do brakes have to absorb? Energy answers these without tracking every instant, and conservation of energy is one of the most reliable rules in all of physics.

Work

When a force pushes an object through a displacement, it does work:

W = F · d · cos θ

F is the force, d is the distance moved, and θ is the angle between the force and the direction of motion. Work is measured in joules (J), and 1 J = 1 N·m.

• Pushing a cart 5 m with 20 N along the direction of motion: W = 20 × 5 = 100 J.
• Pulling a sled with 20 N on a rope at 60° above the ground for 5 m: W = 20 × 5 × cos 60° = 50 J. Only the part of the force along the motion does work.
• A force perpendicular to the motion does no work (cos 90° = 0). The normal force on a car driving on a flat road does no work.
• Friction usually points against the motion (θ = 180°), so it does negative work: it takes energy out of the motion and turns it into heat.

Kinetic and potential energy

Kinetic energy is the energy of motion:

KE = ½ · m · v²

A 1,000 kg car at 20 m/s has KE = ½ × 1000 × 20² = 200,000 J = 200 kJ. Because v is squared, the same car at 40 m/s has 800 kJ: four times as much energy for the brakes to absorb.

Gravitational potential energy is energy stored by height:

PE = m · g · h

h is height above a reference level you choose. Only changes in height matter, so pick whatever reference is convenient (often the lowest point).

A stretched or compressed spring also stores energy, ½·k·x², where k is the spring constant and x the stretch.

The work-energy theorem

The net work done on an object equals its change in kinetic energy:

W_net = ΔKE = ½·m·v² − ½·m·v₀²

This is Newton's second law rewritten in energy terms. It is often the fastest route to a speed.

Conservation of energy

If only gravity (and springs) do work, the total mechanical energy stays constant:

KE₀ + PE₀ = KE + PE

When friction or air drag act, some energy leaves the motion as heat. Energy is still conserved overall; we just have to account for it:

KE₀ + PE₀ = KE + PE + W_friction

where W_friction is the energy lost to friction (a positive number).

Worked example 1: a skateboarder

A 60 kg skateboarder starts from rest at the top of a ramp 4.0 m high. Ignoring friction, how fast is she going at the bottom?

m·g·h = ½·m·v²     →     v = √(2·g·h) = √(2 × 9.81 × 4.0) = 8.86 m/s

The mass cancels: a heavy and a light skater reach the same speed (without friction). The shape of the ramp does not matter either, only the height.

Worked example 2: with friction

Now suppose friction and air drag take 500 J out of her motion on the way down.

PE at top      = 60 × 9.81 × 4.0  = 2354 J
energy left    = 2354 − 500       = 1854 J  (all kinetic at the bottom)
v              = √(2 × 1854 / 60) = 7.86 m/s

Power

Power is the rate of doing work, measured in watts (W): 1 W = 1 J/s.

P = W / t

Lifting a 50 kg load 3.0 m takes W = 50 × 9.81 × 3.0 = 1,472 J. Doing it in 10 s needs 147 W; doing it in 2 s needs 736 W. The work is the same; the power is not. Motors, engines and people are rated by power. Real machines are less than 100% efficient, so the input power is always larger than the useful output.

In code

import math

def speed_at_bottom(height, g=9.81):
    return math.sqrt(2 * g * height)

print(round(speed_at_bottom(4.0), 2))   # 8.86

Common mistakes

• Forgetting to square v. KE is ½mv², not ½mv.
• Mixing kJ and J, or grams and kilograms. Convert everything to SI first.
• Treating work as a vector. Work and energy are scalars: they have a size and sign, but no direction.
• Ignoring losses. Real systems lose energy to friction, drag and heat. Frictionless results are an upper bound on speed.

Quick recap

• W = F·d·cos θ, in joules.
• KE = ½mv², PE = mgh.
• Net work = change in KE.
• Total energy is conserved; friction converts some of it to heat.
• Power = work per unit time, in watts.

Try it

In the Do tab, use an energy balance to find the speed at the bottom of a ramp, with and without friction losses.