Kinetic and Potential Energy

Objective: Apply K = 1/2 mv² and compare the effects of speed and mass.

Translational kinetic energy in the Newtonian model is K = 1/2 mv², where m is mass in kilograms and v is speed in meters per second; the result is joules. A 2 kg object moving at 3 m/s has 9 J of translational kinetic energy. Doubling speed at fixed mass multiplies K by four; doubling mass at fixed speed doubles K. Kinetic energy is nonnegative and does not indicate motion direction, unlike velocity. The formula treats the object’s overall translational motion and does not by itself include rotational, thermal, or internal energy. Speed estimates and units matter because squaring magnifies numerical errors.

Objective: Use ΔU = mgΔh near Earth with a reference level.

Near Earth’s surface, where g is approximately constant, change in gravitational potential energy of an object–Earth system is ΔU_g = mgΔh. Raising a 2 kg object by 5 m changes U by about +98 J when g = 9.8 N/kg. The expression depends on vertical height change, not path length up a winding ramp. Lowering the object makes ΔU negative relative to the same sign convention. Potential energy belongs to an interaction and reference choice, rather than being an intrinsic material stored in an isolated object. At larger altitude changes, constant g becomes a less accurate approximation.

Objective: Use a spring model and state when it applies.

For an ideal spring obeying Hooke’s law within its elastic range, restoring force magnitude is F = k|x| and elastic potential energy is U_s = 1/2 kx² relative to its relaxed length. The spring constant k has units N/m, and x is compression or extension in meters. A spring with k = 200 N/m displaced 0.10 m stores 1 J in this model. Doubling displacement quadruples U_s while doubling k at fixed displacement doubles it. Real springs can depart from linear behavior or become permanently deformed; the simple formula should not be extrapolated beyond the tested elastic range.