Kinetic and Potential Energy Calculations

Kinetic energy E_k = ½mv², gravitational potential energy E_p = mgh and elastic potential energy E_e = ½kx² are the three energy formulas of mechanics, and they are used to calculate speeds, heights and compressions when energy changes from one form to another. In brief: in each formula the quantity to watch is squared (v and x) or linear (h). Use SI units throughout (kg, m, s, N/m) to get joules. Changes of energy, such as ΔE_p = mgΔh, are more important than absolute values.

This guide gives the three formulas with their meaning and conditions of use, shows a systematic method for problems, compares the formulas in a table, works through a series of calculations on a baseball, a mountaineer, a spring and a falling object, and corrects common mistakes. The basic concepts of kinetic and potential energy were introduced in an earlier article; here the emphasis is on calculation.

What are the formulas?

Kinetic energy: E_k = ½mv². It depends on the mass (linearly) and the speed (squared). Rearranged: v = √(2E_k ÷ m).
Gravitational potential energy (near Earth): E_p = mgh, relative to a reference level where h = 0. The change is ΔE_p = mgΔh. Valid when g is constant, that is, for heights small compared with the radius of the Earth (6,371 km).
Elastic potential energy of a spring: E_e = ½kx², where x is the extension or compression and k the spring constant. It is the area under the force-extension graph.

How should problems be solved?

Choose the reference level for height (usually the lowest point).
List the data in SI units.
Identify which formula applies to each stage, and compute energy.
Compare with the expected size, and state the unit (J).

Elastic energy E = ½kx² for the spring with k = 400 N/m. Compressing it by 0.10 m stores 2.0 J, and doubling the compression to 0.20 m stores 8.0 J, four times as much. The curve is a parabola because the energy depends on the square of x.

The formulas compared

Mechanical energy formulas and what happens when a variable doubles
Energy	Formula	Doubling the variable
Kinetic	½mv²	Mass ×2: energy ×2; speed ×2: energy ×4
Gravitational potential	mgh	Mass ×2 or height ×2: energy ×2
Elastic potential	½kx²	k ×2: energy ×2; x ×2: energy ×4

Worked example: calculations

Baseball. A 0.145 kg ball is thrown at 40 m/s. E_k = ½ × 0.145 × 40² = 0.0725 × 1,600 = 116 J.
Speed from energy. A 1,200 kg car has 375,000 J of kinetic energy. v = √(2 × 375,000 ÷ 1,200) = √625 = 25 m/s.
Mountaineer. A climber of mass 60 kg ascends from 500 m to 2,500 m above sea level. ΔE_p = 60 × 9.8 × 2,000 = 1.18 × 10⁶ J = 1.18 MJ, about the energy in 280 kcal of food (1.18 × 10⁶ ÷ 4,184), though the body is only about 20 to 25 percent efficient, so the climber uses more than 1,100 kcal.
Spring. A spring with k = 400 N/m is compressed by 0.10 m. E_e = ½ × 400 × 0.10² = 2.0 J. If compressed by 0.20 m, the energy is 8.0 J (four times as much).
Combined. The compressed spring (2.0 J) launches a 0.10 kg ball straight up. If all the energy is transferred to the ball, v = √(2 × 2.0 ÷ 0.10) = √40 = 6.3 m/s at launch, and the ball rises to a height h = E ÷ (mg) = 2.0 ÷ (0.10 × 9.8) = 2.04 m.

How did the formulas arise?

The kinetic energy formula follows from Leibniz’s vis viva (mv²) in the 1680s. The factor ½ was introduced by Gaspard-Gustave Coriolis in 1829, to make the kinetic energy equal to the work done by a force in producing it. Hooke’s law (1678) leads to the elastic energy formula, and Newton’s gravitation to the expression for gravitational potential energy.

Which mistakes are common?

Mistake 1: Not squaring the speed or the extension

Both kinetic and elastic energy involve squares.

Mistake 2: Using grams or centimeters

Convert to kg and m before using the formulas.

Mistake 3: Forgetting that potential energy needs a reference level

State where h = 0. Only changes in energy are measurable.

Mistake 4: Using the slant length instead of the vertical height

For gravitational potential energy, use the vertical change.