Push a stalled car and it slowly starts rolling — you've done work on it, and that work shows up as motion. Lift a backpack onto a high shelf and you've stored something: let go and it falls, handing back exactly what you put in. That stored-up "ability to make things move or change" is what we call energy. Work is the act of transferring it by pushing through a distance, and power is simply how fast you transfer it — sprinting up the stairs and strolling up use the same energy, but sprinting demands far more power.
Putting numbers on it
Work is force multiplied by the distance you push it: $W = Fd$. Shove a box with 10 N over 3 m and you've done $30$ joules of work. That work turns into kinetic energy, $KE = \tfrac{1}{2}mv^2$ — the energy of moving — or gets banked as potential energy: $PE = mgh$ for height, or $PE = \tfrac{1}{2}kx^2$ for a stretched spring. Lift a $2\text{ kg}$ book up $1.5\text{ m}$ and you store $mgh = 2 \times 9.81 \times 1.5 \approx 29\text{ J}$; drop it and those same 29 J reappear as speed at the bottom.
The precise version
Only the part of the force pointing along the motion counts, so in general $W = Fd\cos\theta$ — a force at right angles to the path (like the normal force) does zero work. The work–energy theorem ties it all together: the net work equals the change in kinetic energy, $W_{\text{net}} = \Delta KE$. When only gravity and springs act, the total $KE + PE$ stays constant — energy just trades form. Power is the time-rate of work, $P = \dfrac{dW}{dt} = Fv$.
The whole idea in one line
$$W_{\text{net}} = \Delta KE, \qquad KE + PE = \text{const (no friction)}, \qquad P = Fv$$
In the simulation above, the Force F and Mass m sliders set how quickly KE climbs, Friction μ bleeds mechanical energy off into heat, Spring k and x₀ set how much elastic PE you start with, and Gravity g scales the $mgh$ stored on the ramp.
Try this in the sim above
Open Spring-Mass and set $\mu = 0$: KE and PE swap back and forth forever while the green total-energy line stays perfectly flat — conservation made visible. Now switch to Block on Surface and raise $\mu$: watch the total-energy line sag downward as motion drains into heat. Finally open Power Engine and notice the driving force shrinking as the speed grows — that is $P = Fv$ holding the power constant while $F$ and $v$ trade off.
Work–Energy Theorem. From Newton's 2nd Law: $F=ma$. Multiply both sides by displacement $dx$ and integrate: $\int F\,dx = m\int a\,dx = m\int v\,dv = \tfrac{1}{2}mv_f^2 - \tfrac{1}{2}mv_i^2$. Therefore $W_{\text{net}} = \Delta KE$.
2
Conservative forces & PE. A force is conservative if $W = -\Delta PE$. Gravity: $W_g = mgh_i - mgh_f = -\Delta PE_g$. Spring: $W_s = \tfrac{1}{2}kx_i^2 - \tfrac{1}{2}kx_f^2 = -\Delta PE_s$.
3
Mechanical energy conservation. If only conservative forces act: $W_{\text{net}} = -\Delta PE = \Delta KE$, so $\Delta(KE + PE) = 0$, i.e. $E_{\text{mech}} = \text{const}$.
4
Energy dissipation by friction. Friction is non-conservative: $W_f = -\mu_k mg\,d < 0$. Mechanical energy decreases by $|W_f|$, converted to thermal energy: $\Delta E_{\text{thermal}} = \mu_k mg\,d$.
5
Spring-mass energy. At extension $x$: $PE_s = \tfrac{1}{2}kx^2$. At equilibrium ($x=0$): all PE converts to KE. Max speed: $v_{\max} = x_0\sqrt{k/m}$ (from $\tfrac{1}{2}kx_0^2 = \tfrac{1}{2}mv_{\max}^2$).
6
Power. $P = dW/dt = F\,dx/dt = Fv$. For constant force: average power $\bar{P} = W/t$. Engine efficiency: $\eta = P_{\text{out}}/P_{\text{in}} \leq 1$.
Block on Surface: A horizontal force moves a block against friction — watch KE rise and total mechanical energy decrease due to friction losses. Spring-Mass: A compressed spring releases, converting PE to KE and back — total energy stays constant with no friction. Ramp & Loop: A ball rolls down a ramp and through a loop — gravitational PE converts to KE; you can check whether it completes the loop. Power Engine: Visualises how an engine delivers constant power — velocity and force change over time while $P = Fv = \text{const}$.
Key takeaway: Every mode illustrates one aspect of the same principle — energy transforms between forms but the total is conserved when friction is absent.
Only when force and displacement are parallel. In general, $W = Fd\cos\theta$, where $\theta$ is the angle between force and displacement. A force perpendicular to motion does zero work — for example, the normal force on a horizontal surface, or the centripetal force in circular motion. This is why a person carrying a bag horizontally does no work against gravity, even though they may feel tired (muscles do internal work).
Key takeaway: Work is a dot product — direction matters. Zero displacement or 90° angle → zero work, regardless of how large the force is.
Because the spring force is not constant — it increases linearly from 0 to $kx$ as you stretch it. Work is the area under the force-displacement graph. For a linear force $F = kx$, the area under $F$ vs $x$ from 0 to $x$ is a triangle: $W = \tfrac{1}{2} \times base \times height = \tfrac{1}{2} \times x \times kx = \tfrac{1}{2}kx^2$. The factor of $\frac{1}{2}$ is the signature of any quadratic potential from a linear restoring force.
Key takeaway: $PE = \tfrac{1}{2}kx^2$ comes from integrating $F = kx$. The ½ reflects that you "ramp up" the force gradually, not apply it all at once.
Work: Lifting groceries, pushing a car, compressing a gas piston. Energy: Food calories (chemical PE → body heat + mechanical KE), battery charge (electrical PE), compressed springs in watches (elastic PE). Power: A 100 W bulb consumes 100 J every second. A car engine rated at 150 kW can deliver 150,000 J per second. A human cyclist produces about 200–400 W sustained. Hydroelectric dams convert gravitational PE of water to electrical energy at gigawatt scales.
Key takeaway: Energy is the "what" (how much capacity), power is the "how fast" — the same total work done slowly or quickly requires the same energy but very different power.
Friction converts macroscopic kinetic energy into microscopic thermal energy — the random vibrations of atoms in the surfaces. This energy is not "lost" — it heats the materials. Total energy (KE + PE + thermal) is always conserved; only mechanical energy decreases. This is the 1st Law of Thermodynamics in disguise. You can feel this with your hands: rub them together and they warm up. The simulation shows mechanical energy decreasing when friction is nonzero — the "missing" energy is heat.
Key takeaway: "Lost to friction" means converted to heat, not destroyed. The universe's total energy is always constant.
A single object cannot have KE without momentum ($KE = p^2/2m$, so $KE=0 \Leftrightarrow p=0$ for a single particle). However, a system of objects can have zero total momentum but nonzero total KE — for example, two equal masses moving in opposite directions. Conversely, an object at rest has zero KE and zero momentum simultaneously. This distinction matters in collisions: elastic collisions conserve both; inelastic ones conserve only momentum.
Key takeaway: For a single object, KE and momentum are linked by $KE = p^2/2m$. For systems, they can be independently zero or nonzero.
Resources: Khan Academy — Work & Energy (khanacademy.org/physics); HyperPhysics — Work, Energy, Power (hyperphysics.phy-astr.gsu.edu); MIT OCW 8.01 — Lectures 11–13.
Section 05
Common Misconceptions
❌ A force always does work on an object it acts upon.
✅ Work requires both a force and displacement in the direction of that force. A wall exerts a force on a leaning person but does zero work (no displacement). The normal force on a moving object on a flat surface does zero work (perpendicular to motion). Centripetal force in circular motion does zero work — speed and KE stay constant even though force is nonzero.
📖 Halliday, Resnick & Walker, 10th Ed., §7-2: Work Done by a Constant Force.
❌ More energy means more power.
✅ Power is the rate of energy transfer, not the amount. A 1 W motor running for 1000 seconds delivers 1000 J — more energy than a 500 W motor running for 1 second (500 J). But the 500 W motor is far more powerful. A slow human climber and a fast elevator motor can deliver the same total energy to raise a load; the motor is far more powerful because it does it much faster.
📖 Serway & Jewett, 8th Ed., §8.5: Power.
❌ If energy is conserved, friction doesn't really waste anything.
✅ Total energy is conserved, but useful mechanical energy is lost to heat which cannot easily be recovered. This is the 2nd Law of Thermodynamics: heat flows spontaneously from hot to cold, and converting heat back to mechanical work is always incomplete. So while no energy is destroyed, friction degrades it into a less useful form. Engineering efficiency losses due to friction are real and costly.
✅ Gravitational PE is a property of the object-Earth system, not just the object. Similarly, spring PE is stored in the spring (the field between the coils), not in the attached mass. This distinction matters in advanced physics: field energy (electromagnetic, gravitational) is distributed through space, not localised in particles. The choice of reference point for PE is arbitrary — only changes in PE have physical meaning.
❌ In the spring-mass system, PE is maximum when the spring is at natural length.
✅ Spring PE is zero at natural length (equilibrium) and maximum at maximum displacement. Gravitational PE for a vertical spring-mass has its own reference point. The total PE minimum (equilibrium position) is where KE is maximum and speed is greatest. Students often confuse the equilibrium position with a maximum-energy state because the restoring force is zero there — but zero force ≠ zero energy.