Kinetic/Potential Energy, Conservation of Energy & Power

Kinetic Energy

Kinetic energy (KE) is the energy of motion:

KE = ½mv²

where m is mass (kg) and v is speed (m/s). Like all forms of mechanical energy, KE is measured in joules (J), and 1 J = 1 kg·m²/s². Because speed is squared, kinetic energy is extremely sensitive to changes in speed: doubling an object's speed quadruples its kinetic energy, and tripling its speed increases KE ninefold. This nonlinear, quadratic scaling is a favorite MCAT contrast with linear momentum, p = mv, which scales directly with speed — a car going twice as fast has twice the momentum but four times the kinetic energy, and consequently needs roughly four times the stopping distance under a constant braking force (from W_net = ΔKE with friction work proportional to distance).

Worked example: A 70 kg person jogs at 2 m/s: KE = ½(70)(4) = 140 J. At 4 m/s, KE = ½(70)(16) = 560 J — four times larger for only twice the speed.

Potential Energy

Potential energy (PE) is stored energy associated with an object's position or configuration, available to convert into kinetic energy under the action of a conservative force. The MCAT tests two specific forms.

Gravitational potential energy, near Earth's surface where gravitational field strength g is essentially constant:

PE = mgh

where h is height measured from a chosen reference level. The reference level, and therefore the "zero" of PE, is arbitrary — only changes in PE have physical meaning. Climbing a 0.5 m step with mass 60 kg (g ≈ 10 m/s²) raises PE by (60)(10)(0.5) = 300 J per step.

Elastic (spring) potential energy, stored in a compressed or stretched ideal spring:

PE = ½kx²

where k is the spring constant (N/m, a measure of stiffness) and x is the displacement from the spring's natural, unstretched equilibrium length. Because x is squared, a spring compressed 0.05 m stores exactly the same potential energy as one stretched 0.05 m. Spring force follows the related relationship F = −kx (Hooke's Law), where the negative sign shows the force always points back toward equilibrium — but PE itself is always positive (or zero exactly at equilibrium), since x² cannot be negative. Tendons and some ligaments behave approximately like nonlinear springs; MCAT passages may model them with an effective k for small displacements.

Conservation of Mechanical Energy

When only conservative forces act on a system — gravity, spring force, with no friction, air resistance, or applied muscular force — total mechanical energy, the sum of kinetic and potential energy, is conserved:

KE_i + PE_i = KE_f + PE_f

This single equation is one of the most powerful problem-solving tools on the MCAT because it relates speed at one point directly to height or spring compression at another point, without ever needing the elapsed time or the exact path taken. A ball dropped from a height, a pendulum swinging, and a ball launched by a spring are all classic conservation-of-energy scenarios. Once friction, air resistance, or an external applied force enters the picture, some mechanical energy is lost to heat (or gained from outside work), and the more general relationship becomes W_nc = ΔKE + ΔPE, where W_nc is the work done by non-conservative forces.

In living systems, pure mechanical-energy conservation is almost never exact: muscle does positive or negative work, and friction in joints and soft tissue dissipates energy. Still, short aerial phases of gait, projectile motion of a thrown object, or a frictionless lab model of a joint often allow the ideal conservation equation as a first approximation.

Power

Power (P) is the rate at which energy is transferred or work is done:

P = W/t (average power)

Power is measured in watts (W), where 1 W = 1 J/s = 1 kg·m²/s³. For a constant force acting parallel to an object's velocity, instantaneous power can also be written as P = Fv — useful for describing an engine, motor, or muscle sustaining a given force at a given speed. Two systems can do the identical total amount of work yet require very different power outputs depending on how quickly that work is performed: a person who climbs a flight of stairs in 5 seconds does the same work against gravity as one who climbs the identical stairs in 30 seconds, but the first person's body generates roughly six times the power.

Energy and power formulas at a glance:

Worked Example: Spring-Launched Ball and Average Power

A physics-of-motion lab uses a horizontal spring (spring constant k = 200 N/m) to launch a 0.50 kg ball along a frictionless track that curves upward into a frictionless ramp. A technician compresses the spring by x = 0.10 m and releases it.

Step 1 — Elastic potential energy stored in the compressed spring.

PE_spring = ½kx² = ½(200 N/m)(0.10 m)² = ½(200)(0.01) = 1.0 J

Step 2 — Speed of the ball as it leaves the spring.

On the frictionless track, all of the spring's potential energy converts to kinetic energy:

PE_spring = KE_ball → 1.0 J = ½(0.50 kg)v² → v² = 4.0 m²/s² → v = 2.0 m/s

Step 3 — Maximum height on the ramp.

As the ball rises, kinetic energy converts to gravitational potential energy. At the ball's highest point, all 1.0 J of mechanical energy is stored as PE_grav, since its velocity is momentarily zero. Using g ≈ 10 m/s²:

mgh = 1.0 J → h = 1.0 / [(0.50)(10)] = 1.0/5.0 = 0.20 m

Because the track and ramp are frictionless, total mechanical energy is the same 1.0 J at every point in the motion — only its form (elastic, then kinetic, then gravitational) changes. If the ramp had friction, some of that 1.0 J would be lost as heat, and the ball would reach a lower maximum height than energy conservation alone predicts.

Step 4 — Average power to compress the spring.

Suppose the technician takes 0.20 s to compress the spring the 0.10 m needed to store that 1.0 J of elastic potential energy. The average power delivered is:

P_avg = W/t = 1.0 J / 0.20 s = 5.0 W

Walking power estimate: A 70 kg person climbs a 3 m staircase in 6 s. Work against gravity ≈ mgh = (70)(10)(3) = 2,100 J; average power ≈ 2,100/6 = 350 W during the climb. Resting basal metabolism is on the order of 100 W continuous, so brief stair climbing is a several-fold increase in power demand — most of which, as vertebrate physiology covered elsewhere in this guide explains, is ultimately released as heat rather than external mechanical work.