Kinetic & Potential Energy Calculator (AP Physics 1)

What the calculator above computes

The calculator above handles the three energy formulas at the heart of AP Physics 1 Unit 3: translational kinetic energy $K = \frac{1}{2}mv^2$, change in gravitational potential energy $\Delta U_g = mg\Delta y$, and elastic (spring) potential energy $U_s = \frac{1}{2}k(\Delta x)^2$. Pick the quantity you need, enter the known values in SI units, and the result comes out in joules.

These three expressions are the building blocks of every energy bar chart and conservation of energy problem you will see this year. A block sliding down a ramp trades $\Delta U_g$ for $K$. A dart gun trades $U_s$ for $K$. Getting fast and accurate with each formula on its own is the first step toward chaining them together in a full problem.

Kinetic energy depends on speed squared

Translational kinetic energy is the energy an object has because it is moving:

Mass $m$ goes in kilograms and speed $v$ in meters per second. Two features matter on the AP exam. First, $K$ scales linearly with mass but with the square of speed: doubling the mass doubles the kinetic energy, while doubling the speed quadruples it. That squared dependence is why highway crashes are so much worse than parking-lot crashes, and why braking distance grows faster than speed does. Second, $K$ is a scalar that can never be negative. Direction is irrelevant: a ball moving at 5 m/s east has exactly the same kinetic energy as one moving at 5 m/s west, because $K$ depends on $v^2$, so the sign of the velocity drops out. This is Topic 3.1 in the CED, and it pairs directly with the work-energy theorem.

Gravitational potential energy: only changes matter

Near Earth's surface, the AP Physics 1 equation sheet writes gravitational potential energy as a change:

with $g = 9.8\ \mathrm{m/s^2}$. The deltas are deliberate. There is no single correct potential energy at a point, because you are free to call any height zero. Only the change in height between two positions has physical meaning, which is why the calculator above asks for a height change $\Delta y$ rather than an absolute height. If the object rises, $\Delta y$ and $\Delta U_g$ are both positive: the system gains stored energy. If it falls, both are negative, and the lost potential energy usually reappears as kinetic energy. One wording detail AP graders reward: potential energy belongs to the object-Earth system, not to the object by itself. A lone object cannot have gravitational potential energy.

Spring potential energy: stretch or compress, energy is stored

A spring that obeys Hooke's law, $F_s = -k\Delta x$, stores elastic potential energy

where $k$ is the spring constant in newtons per meter and $\Delta x$ is the stretch or compression measured from the spring's natural (relaxed) length, in meters. Because $\Delta x$ is squared, the stored energy is positive whether you stretch or compress: 5 cm of compression stores exactly as much energy as 5 cm of stretch. The square also means storage is not proportional to displacement. Double the compression and you store four times the energy. The most common error is measuring $\Delta x$ from the wrong place: it is always the distance from the relaxed length, not from wherever the block happens to start. Spring energy comes back in Unit 7, where trading $U_s$ for $K$ drives simple harmonic motion.

What a joule actually is

Every result from the calculator above comes out in joules (J), the SI unit of energy. One joule is the work done when a force of one newton pushes through one meter:

To get a feel for the size, lift a 0.10 kg apple through 1.0 m: that stores $(0.10)(9.8)(1.0) = 0.98$ J, almost exactly one joule. A 60 kg sprinter at 8.0 m/s carries $\frac{1}{2}(60)(8.0)^2 = 1920$ J. One food Calorie is 4184 J. Joules only appear if everything goes in as SI units: kilograms, meters per second, meters, and newtons per meter. Convert grams to kilograms and centimeters to meters before substituting, and the units take care of themselves.

Where these formulas lead on the AP exam

Unit 3 (Work, Energy, and Power) carries 18 to 23 percent of the AP Physics 1 multiple-choice section, and it rarely tests these formulas in isolation. The work-energy theorem connects work to kinetic energy: the net work done on an object equals its change in $K$. Conservation of energy chains all three quantities together: for a system with no friction or external work, the total of $K$, $U_g$, and $U_s$ stays constant, so energy just shifts between the three buckets this page computes. Kinetic energy also decides how a collision is classified, since it is conserved only in elastic ones, the comparison that drives conservation of momentum problems. Power, covered by the work and power calculator, measures how fast that energy transfers: $P_{avg} = W/\Delta t$. A reliable habit for multi-step problems: compute each energy term separately with the calculator above, then check that the before and after totals balance.