14.2 Energy, Work, and Power

Energy ideas connect almost every USTET Science topic: motion, heat, circuits, and waves. This section focuses on mechanical work and energy—definitions, formulas, conservation, and power—so you can track where energy goes in short word problems.

What Is Energy?

Energy is the capacity to do work or produce change. The SI unit of energy (and of work) is the joule (J): $1\ \mathrm{J} = 1\ \mathrm{N\cdot m} = 1\ \mathrm{kg\cdot m^2/s^2}$. Energy is a scalar: it has magnitude but no direction, though the processes that transfer energy often have direction.

Everyday language says energy is "used up." Physics says energy is transformed from one form to another or transferred between systems. Total energy in an isolated system is conserved (first law of thermodynamics / conservation of energy).

Common Forms of Energy

A motorcycle climbing a hill converts chemical energy in fuel into kinetic energy and gravitational potential energy; some energy also becomes thermal energy through friction and exhaust. Tracking the dominant conversion is usually enough for entrance-exam items.

Work: Force Through a Displacement

Work done by a constant force is:

$W = Fd\cos\theta$

where $F$ is force magnitude, $d$ is displacement magnitude, and $\theta$ is the angle between the force and displacement vectors.

• If force is parallel to displacement ($\theta = 0°$), $W = Fd$.
• If force is perpendicular to displacement ($\theta = 90°$), $W = 0$ (the force does no work).
• If force opposes displacement ($\theta = 180°$), work is negative—the force removes energy from the object's mechanical energy budget.

Carrying a backpack at constant height across a flat hallway, the upward force you apply to support the pack is perpendicular to horizontal displacement, so that supporting force does no work. Pushing a desk across the floor in the direction of motion does positive work.

Worked example — Simple work

You push a box 5.0 m across a rough floor with a horizontal force of 40 N. How much work do you do?

$W = (40\ \mathrm{N})(5.0\ \mathrm{m}) = 200\ \mathrm{J}$

If friction is 15 N opposite the motion, friction does $W_f = -(15)(5.0) = -75\ \mathrm{J}$.

Kinetic Energy

Kinetic energy of a mass $m$ moving at speed $v$ is:

$KE = \frac{1}{2}mv^2$

Because of $v^2$, doubling speed quadruples KE. A 2.0 kg object at 3.0 m/s has

$KE = \frac{1}{2}(2.0)(3.0)^2 = 9.0\ \mathrm{J}$

At 6.0 m/s, KE = 36 J—four times as much.

Gravitational Potential Energy

Near Earth's surface, taking a reference height where $PE = 0$:

$PE = mgh$

Raising a 5.0 kg box by 2.0 m (using $g = 10\ \mathrm{m/s^2}$ for a quick exam estimate) stores

$PE = (5.0)(10)(2.0) = 100\ \mathrm{J}$

Only changes in height matter for $\Delta PE$. The reference level is arbitrary as long as you are consistent within one problem.

Work–Energy Theorem

The work–energy theorem states that the net work done on an object equals its change in kinetic energy:

$W_{net} = \Delta KE = KE_f - KE_i$

If net work is positive, the object speeds up. If net work is negative, it slows down. This links Section 14.1 forces to energy bookkeeping: compute $W_{net}$ from all forces, or compute $\Delta KE$ from speeds—they must match.

Worked example — Work–energy

A 3.0 kg cart is at rest. A net force does 54 J of work on it. What is its final speed?

$W_{net} = \Delta KE = \frac{1}{2}mv_f^2 - 0$ $54 = \frac{1}{2}(3.0)v_f^2$ $v_f^2 = 36 \Rightarrow v_f = 6.0\ \mathrm{m/s}$

Conservation of Mechanical Energy

Mechanical energy $E = KE + PE$ (plus elastic PE if springs matter). When only conservative forces (gravity, ideal springs) do work—or when non-conservative work is negligible—mechanical energy is conserved:

$KE_i + PE_i = KE_f + PE_f$

Worked example — Falling object

A 2.0 kg ball is dropped from rest at height 5.0 m. Using $g = 10\ \mathrm{m/s^2}$, find its speed just before hitting the ground (ignore air resistance).

Take $PE = 0$ at the ground. Initially: $KE_i = 0$, $PE_i = mgh = 100\ \mathrm{J}$. Finally: $PE_f = 0$, so $KE_f = 100\ \mathrm{J}$.

$\frac{1}{2}(2.0)v^2 = 100 \Rightarrow v^2 = 100 \Rightarrow v = 10\ \mathrm{m/s}$

Same result as free-fall kinematics with $v^2 = 2gh$. Energy methods shine when path shapes are awkward but heights are known.

When friction or air drag matters, mechanical energy decreases: the "missing" energy becomes thermal energy. Then:

$KE_i + PE_i + W_{nc} = KE_f + PE_f$

where $W_{nc}$ is work by non-conservative forces (often negative).

Power

Power is the rate of energy transfer or work:

$P = \frac{W}{t} = \frac{E}{t}$

The SI unit is the watt (W): $1\ \mathrm{W} = 1\ \mathrm{J/s}$. (Do not confuse the unit symbol W with the variable for work—context distinguishes them.) For a force parallel to constant velocity:

$P = Fv$

Worked example — Power

A motor does 1,500 J of work in 5.0 s. What is its average power?

$P = \frac{1500\ \mathrm{J}}{5.0\ \mathrm{s}} = 300\ \mathrm{W}$

A student pushes with 50 N while moving at a steady 2.0 m/s in the direction of the force: $P = Fv = 100\ \mathrm{W}$.

Efficiency (Light Touch)

Real machines waste some input energy as heat and sound. Efficiency is useful output energy (or work) divided by input energy, often expressed as a percent. An efficiency of 40% means 0.40 of the input becomes desired output; the rest is transformed into less useful forms. USTET may ask qualitative "where did the energy go?" rather than deep efficiency algebra.

Strategy for Word Problems

1. Identify what is asked: work, KE, PE, speed, height, or power.
2. List knowns with units; choose $g = 9.8$ or $10$ as the problem implies.
3. Decide whether mechanical energy is conserved or whether net work / friction must appear.
4. Write one governing equation ($W = Fd$, $KE = \tfrac{1}{2}mv^2$, $PE = mgh$, $W_{net} = \Delta KE$, or $P = W/t$) and solve.
5. Sanity-check: heavier and faster means more KE; higher means more PE; same work in less time means more power.

Energy problems reward clear bookkeeping. Name the form at the start, name the form at the end, and account for heat when friction is mentioned—those three habits cover most USTET energy items.