9,810
J
9.81
kJ
9,810
J
981
J/kg
9,810
J
9.81
kJ
9,810
J
981
J/kg
Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. Two formulations exist depending on the context:
Near-surface approximation: $$PE = mgh$$
Valid when the height $$h$$ is small compared to the body's radius, so $$g$$ is approximately constant. This is the formula used in everyday physics — dropping a ball, lifting a weight, or analyzing roller coasters.
Universal formula: $$PE = -\frac{GMm}{r}$$
Valid at any distance, with the zero reference at infinity. The negative sign indicates that the object is gravitationally bound — you must add energy to move it to infinity. This is the formula used in orbital mechanics, escape velocity derivations, and astrophysics.
Our calculator supports both modes. The near-surface mode is ideal for engineering and everyday physics problems, while the universal mode handles satellite orbits, interplanetary trajectories, and any scenario where $$g$$ varies significantly with position.
Near-surface mode (PE = mgh): Multiply the object's mass by local gravitational acceleration and height. This assumes constant $$g$$ — accurate to within 1% for heights up to about 64 km on Earth.
Universal mode (PE = -GMm/r): Uses Newton's gravitational theory with the convention that $$PE = 0$$ at infinite separation. At finite distance $$r$$, the potential energy is negative, reflecting the energy deficit compared to a free (unbound) particle. The binding energy is $$|PE| = GMm/r$$.
The two formulas connect as follows: for small height $$h$$ above a surface at radius $$R$$, the change in universal PE approximates $$\Delta PE \approx mgh$$, where $$g = GM/R^2$$. The near-surface formula is thus a linearization of the universal formula.
In near-surface mode, the result is always positive (energy stored by lifting the object). In universal mode, the result is always negative (bound state). More negative values mean the object is more tightly bound. To unbind the object (send it to infinity), you must supply energy equal to $$|PE|$$. For a 1000 kg satellite at Earth's surface, the binding energy is about $$6.25 \times 10^{10}$$ J — equivalent to about 15 tons of TNT.
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Lifting 10 kg to 100 meters stores 9,810 J (9.81 kJ) of gravitational potential energy — equivalent to the kinetic energy of a 70 kg person running at about 17 m/s.
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The ISS (mass ~420,000 kg) at orbital radius 6,771 km has a gravitational PE of about -2.47 × 10¹³ J. This enormous binding energy is why de-orbiting the ISS requires careful planning.
By convention, PE is zero when objects are infinitely far apart (unbound). At any finite separation, gravity has done work pulling them together, so the system has less energy than the reference state — hence negative. To separate them back to infinity requires adding energy equal to $$|PE|$$.
Use $$mgh$$ for problems near a surface where $$g$$ is approximately constant — building heights, roller coasters, waterfalls, etc. Use $$-GMm/r$$ when dealing with orbits, escape velocity, or heights comparable to the body's radius.
At escape velocity, the kinetic energy $$\frac{1}{2}mv^2$$ exactly equals the magnitude of gravitational PE $$GMm/r$$. Setting their sum to zero gives $$v_{\text{esc}} = \sqrt{2GM/r}$$.
No. Gravity is a conservative force, meaning the work done depends only on the initial and final positions, not the path. Whether you lift an object straight up or along a winding staircase, the change in PE is the same.
Using the shell theorem and integrating, the potential energy at Earth's center is more negative than at the surface. The gravitational potential at the center of a uniform sphere is $$-3GM/(2R)$$, which is 1.5 times the surface value.
Yes. Hydroelectric dams convert the GPE of elevated water into electricity. Falling objects convert GPE to kinetic energy. In space, gravitational slingshots convert a planet's GPE into spacecraft kinetic energy for interplanetary missions.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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