Choose a gravitational energy reference

Move the zero label without moving the particle

A 2 kg particle starts from rest at fixed height 4 m on a frictionless ramp. Select current height 1 m. The particle has descended 3 m, whatever height we choose to call “zero potential energy.” Moving that accounting reference does not lift the ramp, change gravity or give the particle a push.

The diagram uses a fixed height coordinate. The height control chooses the particle’s physical position on its descent; the reference control moves only the dashed zero-potential line. Compare saved states with the same current height before interpreting a change in an energy bar as a change in the motion.

We use teaching gravitational acceleration g = 10 m/s² and fixed mass m = 2 kg. The gravitational potential energy of the particle–Earth interaction is U = mg(h − href). Here h is the current height in metres and href is the height chosen to have U = 0. U is measured in joules. The product mg is 20 N in this example.

This formula approximates a nearly uniform gravitational field over a small height range. It is not the general gravitational potential for large changes in distance from Earth. The model also neglects Earth’s tiny recoil and represents the particle as a point mass, with no rotational energy.

Calculate three descriptions of one state

At current height 1 m, choose reference height 0 m. Then U = 20 × (1 − 0) = 20 J. The initial potential at height 4 m is 80 J. The change is current minus initial: ΔU = 20 − 80 = −60 J.

Move the reference to height 1 m. The current potential becomes 0 J, and the initial potential becomes 60 J. Their difference is still −60 J. Move the reference again to height 2 m: current potential is −20 J, initial potential is 40 J, and the difference remains −60 J.

All three descriptions predict the same gravity work, Wgravity = −ΔU = +60 J. The particle starts with zero kinetic energy, so it has K = 60 J at height 1 m. Its speed is √(2K/m) = √60 ≈ 7.746 m/s in each description.

A reference shift adds the same constant to potential energy at every position. Algebra makes the cancellation explicit: ΔU = mg(hfinal − href) − mg(hinitial − href) = mg(hfinal − hinitial). The reference terms cancel. Only the height difference remains in the gravity-work calculation.

Negative potential energy is not negative kinetic energy

At height 1 m with reference height 2 m, U = −20 J. This is a valid potential-energy value because the selected state lies below the chosen zero. It does not represent an inaccessible position, a negative speed, or a negative amount of motion energy.

Contrast this with translational kinetic energy K = ½mv². Positive mass and squared speed make K nonnegative. In the previous stopping investigation, a negative candidate K meant the proposed target could not be reached. Applying that conclusion to a negative U would confuse a motion quantity with a quantity whose additive baseline can be chosen.

The energy chart includes negative values on its fixed scale. It does not clip the potential bar at zero or rescale it separately from the other bars. Read the numerical ledger as well as the bar direction: current potential may be negative while current kinetic energy is positive and their sum remains the conserved mechanical energy for that reference.

Keep one reference throughout an equation

A common error is to use an initial potential computed with reference height zero and a final potential computed with reference height two. That mixes two conventions. The resulting difference includes an artificial shift and predicts the wrong speed.

At height 1 m, incorrectly combining initial U = 80 J from the zero-metre reference with final U = −20 J from the two-metre reference would suggest a drop of 100 J. The actual potential decrease is only 60 J. Use the same reference for every term, or explicitly convert all terms together before comparing them.

The OpenStax potential-energy discussion explains the role of a chosen zero. The diagram and values in this lesson are original teaching constructions. A physical height measurement would also require an instrument reference, uncertainty and a stated direction; those measurement issues are separate from the freedom to add a constant to potential energy.

Total energy can shift while conservation remains true

Mechanical energy is E = K + U. With reference height zero, the initial total is 80 J and the total remains 80 J throughout this ideal descent. With reference height two, the total is 40 J throughout. These different totals are consistent descriptions of the same motion.

Conservation compares totals within one consistent convention. It does not require the numerical value of potential energy or total mechanical energy to be identical under all choices of zero. Kinetic energy, speed, elapsed time and potential-energy differences remain unchanged when only the reference is changed.

Transfer: A 3 kg object moves from height 5 m to height 2 m under constant g=10 m/s². Compare initial potential, final potential and gravity work using reference heights 0 m and 4 m. Does changing the reference supply any energy to the object?

Check the potential-reference transfer

With reference zero, the potentials are 150 J and 60 J. With reference 4 m, they are 30 J and −60 J. In both conventions ΔU = −90 J and gravity does +90 J of work. Moving the accounting reference supplies no physical energy and changes no force or motion.