3.3 Work, Energy & Power

Work, energy, and power problems round out the mechanics portion of the NAPT physics content, and two closely related quantities — momentum and density — show up in the same family of problems. All five ideas in this section share one theme: tracking how the capacity to do something moves from one form to another, and how quickly it is delivered.

Work

Work is done on an object when a force causes a displacement in the direction of that force:

W = F × d × cos(θ)

Where F is force (in newtons), d is displacement (in meters), and θ is the angle between the force and the direction of motion. When the force acts in the same direction as the motion (θ = 0°), cos θ = 1 and the formula simplifies to W = F × d. When a force is perpendicular to the motion (θ = 90°), cos θ = 0 and no work is done at all, no matter how large the force is. Work is measured in joules (J); 1 joule equals 1 newton-meter (N·m).

Worked Example 1: Simple Work

A sailor pushes a 200 N crate a horizontal distance of 5 m across the deck, with the push directly in line with the crate's motion.

W = F × d = (200 N)(5 m) = 1,000 J

Worked Example 2: Work at an Angle

A line is used to drag a 150 N fender along the deck for 4 m, with the line held at 60° above horizontal (cos 60° = 0.5).

W = F × d × cos θ = (150)(4)(0.5) = 300 J

Notice that dragging at an angle does less work than pushing the same force in a straight line — part of the pulling force is spent lifting rather than moving the fender forward.

Kinetic and Potential Energy

Energy is the capacity to do work. Two forms matter most for NAPT-level mechanics:

• Kinetic energy (KE) — energy of motion: KE = ½mv²
• Gravitational potential energy (PE) — stored energy due to height: PE = mgh

Where m is mass (kg), v is velocity (m/s), g is 9.8 m/s², and h is height (m) above a chosen reference point. Both are measured in joules, exactly like work.

Worked Example 3: Kinetic Energy

A 4 kg practice shell moves at 10 m/s. Find its kinetic energy.

KE = ½mv² = ½(4)(10²) = ½(4)(100) = 200 J

Worked Example 4: Potential Energy

A 10 kg anchor hangs 5 m above the waterline. Find its gravitational potential energy relative to the water.

PE = mgh = (10)(9.8)(5) = 490 J

Conservation of Energy

The law of conservation of energy states that energy cannot be created or destroyed, only converted from one form into another. In a simple mechanical system with no friction or air resistance, total mechanical energy — kinetic plus potential (KE + PE) — stays constant throughout the motion.

Worked Example 5: A Falling Object Converts Energy

Drop the 10 kg anchor from Worked Example 4 (5 m above the water, starting from rest). Find its velocity just before it hits the water, using conservation of energy instead of the kinematics equations from Section 3.1.

All 490 J of potential energy converts entirely into kinetic energy by the time the anchor reaches the water:

½mv² = 490 J → v² = (2)(490) / 10 = 98 → v = √98 ≈ 9.9 m/s

As a check, the kinematics equation v² = 2gh from Section 3.1 gives the identical result: v² = (2)(9.8)(5) = 98, confirming both methods describe the same physical event.

Power

Power is the rate at which work is done, or energy is transferred, over time:

P = W / t

Where P is measured in watts (W); 1 watt equals 1 joule per second. Power distinguishes how much work gets done from how fast it gets done — the same amount of work performed in less time means more power.

Worked Example 6: Power Output

A davit motor performs 6,000 J of work lifting a small boat in 15 seconds. Find the motor's power output.

P = W / t = 6,000 J / 15 s = 400 W

Momentum

Momentum (p) measures an object's quantity of motion: p = mv (mass times velocity), measured in kilogram-meters per second (kg·m/s). Like velocity, momentum is a vector — direction matters.

The law of conservation of momentum states that in a closed system with no external forces, total momentum before an event (such as a collision) equals total momentum after it.

Worked Example 7: Momentum in a Collision

A 2 kg object moving at 6 m/s collides with a stationary 4 kg object and sticks to it (a perfectly inelastic collision). Find their combined velocity immediately after the collision.

Momentum before = (2 kg)(6 m/s) + (4 kg)(0 m/s) = 12 kg·m/s

Momentum after = (2 kg + 4 kg) × v = (6 kg) × v

Setting them equal: 6v = 12 → v = 2 m/s

Density

Density (ρ) is mass per unit volume:

ρ = m / V

Measured in kilograms per cubic meter (kg/m³) or grams per cubic centimeter (g/cm³). Density is what determines whether an object floats or sinks in a fluid — a hull floats because it displaces a weight of water equal to its own weight, which is only possible because the ship's average density (hull plus the air inside it) is lower than the density of water.

Worked Example 8: Calculating Density

A metal block has a mass of 750 g and a volume of 250 cm³. Find its density.

ρ = m / V = 750 g / 250 cm³ = 3 g/cm³ (equivalent to 3,000 kg/m³)

Since the density of fresh water is 1 g/cm³, this block — at 3 g/cm³ — would sink.

Section Takeaways

• Work equals force times displacement times the cosine of the angle between them (W = Fd cos θ); a force perpendicular to the motion does zero work.
• Kinetic energy (KE = ½mv²) is energy of motion; gravitational potential energy (PE = mgh) is energy of position; both are measured in joules.
• Conservation of energy: with no friction or drag, total mechanical energy (KE + PE) stays constant as an object falls, rises, or swings — only its form changes.
• Power (P = W/t) measures how fast work is done, in watts; momentum (p = mv) measures quantity of motion and is conserved in closed-system collisions.
• Density (ρ = m/V) is mass per unit volume and determines whether an object floats or sinks in a fluid.