Work $W = Fd \cos\theta$, kinetic energy $\frac{1}{2}mv^2$, gravitational potential energy $mgh$, elastic potential energy $\frac{1}{2}kx^2$, conservation of mechanical energy, and power $P = W/t = Fv$

What this dot point is asking

VCAA wants you to apply the concepts of work, energy, and power to mechanical systems, use conservation of energy in problems involving multiple energy types, and analyse situations with and without friction.

Work

Work done by a constant force $F$ acting over a displacement $d$:

where $\theta$ is the angle between force and displacement.

• IMATH_10 (force along motion): $W = F d$ (positive).
• IMATH_12 degrees (perpendicular): $W = 0$. The force does no work.
• IMATH_14 degrees (force opposite motion): $W = -F d$ (negative; the force takes energy away).

Units: joule (J) = N m.

Kinetic energy

The energy of motion:

The work-kinetic energy theorem: the net work on an object equals its change in kinetic energy.

Potential energy

Gravitational PE. $PE_g = mgh$ where $h$ is height above a reference point. The reference is arbitrary; only changes in $PE$ matter.

Elastic PE. A spring with spring constant $k$ stretched or compressed by $x$ from equilibrium: $PE_e = \frac{1}{2} k x^2$.

Conservation of mechanical energy

In an isolated system with only conservative forces (gravity, springs), mechanical energy ($KE + PE$) is conserved:

In practice, friction, air resistance and similar forces dissipate energy as heat:

where $E_{\text{lost}}$ is the work done by friction etc.

Power

Rate of doing work or transferring energy:

Units: watt (W) = J/s.

For an object moving at velocity $v$ with a force $F$ applied:

(For force parallel to motion. More generally $P = \vec{F} \cdot \vec{v}$.)

Worked examples

Falling ball. A 0.5 kg ball is dropped from 10 m. Final speed (ignoring air resistance)?

Conservation: $mgh = \frac{1}{2} m v^2$. $v = \sqrt{2gh} = \sqrt{196} = 14$ m/s.

Spring launch. A 0.2 kg ball is launched by a spring with $k = 100$ N/m compressed by 0.1 m. Maximum speed?

Spring PE: $\frac{1}{2} k x^2 = \frac{1}{2} \times 100 \times 0.01 = 0.5$ J.

Equal to KE at maximum speed: $\frac{1}{2} m v^2 = 0.5$, so $v^2 = 5$, $v \approx 2.24$ m/s.

Power of a car. A 1500 kg car accelerates from 0 to 27 m/s in 5 s. Average power?

$\Delta KE = \frac{1}{2}(1500)(27^2 - 0) = 546,750$ J.

Power = $\Delta KE / t = 546,750 / 5 \approx 109$ kW.

Energy types and conversions

Mechanical (KE + PE) is one form. Others include:

• Thermal (heat).
• Chemical (in fuels).
• Nuclear.
• Electromagnetic (radiation).

Energy can convert between forms. In a falling object with air resistance: gravitational PE -> KE + heat (friction with air).

In a car: chemical PE in fuel -> KE of car + heat (mostly) + sound + light.

Conservation of energy is one of the fundamental laws of physics. In any isolated system, the total energy is constant; only the form changes.

Common errors

Forgetting cosine in work formula. $W = F d \cos\theta$. If force is perpendicular to motion, no work is done.

Confusing PE and KE. PE is positional (height for gravity, displacement for spring). KE is kinetic ($\frac{1}{2} m v^2$).

Mixing $W$ as work and $W$ as weight. Context matters. Work is energy; weight is a force.

Power = work/time, not force. Power is the rate at which work is done.

Friction in conservation problems. When friction is present, mechanical energy is not conserved; account for the energy loss separately.

In one sentence

Work $W = Fd\cos\theta$ done by a force changes an object's kinetic energy ($W_{\text{net}} = \Delta KE$); the total mechanical energy ($KE + PE$) is conserved in systems with only conservative forces (gravity, springs), while friction dissipates energy as heat; power is the rate of energy transfer, $P = W/t = Fv$ for an object moving at velocity $v$ under a parallel force.