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Potential energy for near Earth gravity

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For small height changes, gravitational potential energy can be computed using U g = m g h , {\displaystyle U_{g}=mgh,} where m is the mass in kg, g is the local gravitational field (9.8 metres per second squared on earth), h is the height above a reference level in metres, and U is the energy in joules. In classical physics, gravity exerts a constant downward force F =(0, 0, F z ) on the center of mass of a body moving near the surface of the Earth. The work of gravity on a body moving along a trajectory r (t) = ( x (t), y (t), z (t)), such as the track of a roller coaster is calculated using its velocity, v =( v x , v y , v z ), to obtain W = ∫ t 1 t 2 F ⋅ v d t = ∫ t 1 t 2 F z v z d t = F z Δ z . {\displaystyle W=\int _{t_{1}}^{t_{2}}{\boldsymbol {F}}\cdot {\boldsymbol {v}}\mathrm {d} t=\int _{t_{1}}^{t_{2}}F_{z}v_{z}\mathrm {d} t=F_{z}\Delta z.} where the integral of the vertical...
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Potential energy

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U = ½ · k · x 2(elastic) U = ½ · C · V 2 (electric) U = - m · B (magnetic) In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potential energy of an object that depends on its mass and its distance from the center of mass of another object, the elastic potential energy of an extended spring, and the electric potential energy of an electric charge in an electric field. The unit for energy in the International System of Units (SI) is the joule, which has the symbol J. The term potential energy was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to Greek philosopher Aristotle's concept of potentiality. Potential energy is associated with forces that act on a body in a way that the total work done by these ...
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Potential energy for gravitational forces between two bodies

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The gravitational potential function, also known as gravitational potential energy, is: U = − G M m r , {\displaystyle U=-{\frac {GMm}{r}},} The negative sign follows the convention that work is gained from a loss of potential energy. Derivation The gravitational force between two bodies of mass M and m separated by a distance r is given by Newton's law F = − G M m r 2 r ^ , {\displaystyle \mathbf {F} =-{\frac {GMm}{r^{2}}}\mathbf {\hat {r}} ,} where r ^ {\displaystyle \mathbf {\hat {r}} } is a vector of length 1 pointing from M to m and G is the gravitational constant. Let the mass m move at the velocity v then the work of gravity on this mass as it moves from position r (t 1 ) to r (t 2 ) is given by W = − ∫ r ( t 1 ) r ( t 2 ) G M m r 3 r ⋅ d r = − ∫ t 1 t 2 G M m r 3 r ⋅ v d t . {\displayst...
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