The total energy and mechanical energy have different ranges.
Kinetic energy is the sum of translational energy, K, and potential energy, U. When a mass m moves with a velocity v, then K = ½mv², and in a constant gravitational field, it is represented as U = mgh. The reference height for potential energy can be arbitrarily set, and any physical change is determined by the height difference.
If only restoring force, like gravity, acts, then K + U remains constant. However, if there is friction or inelastic collision, some of the mechanical energy can be converted into internal energy or sound. This does not mean that when mechanical energy decreases, the total energy disappears.
The potential energy associated with free fall and a pendulum transforms into kinetic energy.
For a ball falling from rest through height h without air resistance, mgh = ½mv² gives v = √(2gh). Mass cancels, so the arrival speed is the same for the same fall height. The energy equation does not give elapsed time directly, but quickly compares the initial and final states.
The oscillation has a velocity of 0 at its extremes, and reaches its maximum velocity at the lowest point. An ideal spring exerts no work because its tension is perpendicular to the direction of motion. The actual amplitude decreases because mechanical energy is transferred from the system to surrounding and internal energies.
In circuits and heating applications, we read the mobility using power.
Power P is the energy transferred per unit time. At constant power, E = Pt and 1 W = 1 J/s. A resistor’s power P = VI gives the rate at which electrical energy becomes heat or other forms. If all this energy heats an object without loss, Pt = mcΔT.
We always verify how the boundaries of the system are defined. If we only observe a single object, energy is lost due to friction. However, when we include the object and the surface it’s on, we can calculate an increase in internal energy. The process of maintaining an energy ledger is also helpful in finding losses or measurement errors that may have been missed during experiments.
Key relationships and symbols
K = ½mv² · U = mgh · K + U = constant (only conservative forces do work) · E = Pt
| Symbol | Meaning | Unit |
|---|---|---|
| K, U, E | Kinetic, potential, and transferred energy | J |
| m | mass | kg |
| v | Speed | m/s |
| P | Power | W |
Worked example
If a mass 2 kg object falls from a height 5 m, what happens?
When g = 9.8 m/s², the decreasing gravitational potential energy is 98 J. If there is no friction, the kinetic energy increases by that amount, resulting in a final speed of √98 ≈ 9.90 m/s. However, if friction exists, some of the energy is converted into heat, which can slow it down.
Common misconceptions
- Even in inelastic collisions, where the kinetic energy is reduced, the total energy is not violated.
- Power is the rate of energy transfer, rather than energy itself.
Explore this concept through experiments
Please explain the reason why objects with the same mass, but dropped from the same height, have the same arrival speed, using energy.
One-dimensional collisionsCompare whether the kinetic energy changes even if the coefficient of restitution is changed, such that the amount of motion remains the same.
Pendulum periodObserve the relationship between potential energy and velocity at the extremes and at the lowest point.
Inclined plane and frictionLet's interpret how a portion of mechanical energy is transformed into other forms on a slope with friction.
Specific heat and temperatureConnect the available power and the energy obtained over time to temperature increases.
Ohm’s lawCalculate power from voltage and current, and distinguish the units of power from those of energy.
Electromagnetic inductionWhen creating a induced current, consider the external work required and how it relates to Lenz’s law.