Altitude As Gravitational Potential Energy
An example is the (nearly) uniform gravitational field near the Earth's surface. It has a potential energy
where U is the gravitational potential energy and h is the height above the surface. This means that gravitational potential energy on a contour map is proportional to altitude. On a contour map, the two-dimensional negative gradient of the altitude is a two-dimensional vector field, whose vectors are always perpendicular to the contours and also perpendicular to the direction of gravity. But on the hilly region represented by the contour map, the three-dimensional negative gradient of U always points straight downwards in the direction of gravity; F. However, a ball rolling down a hill cannot move directly downwards due to the normal force of the hill's surface, which cancels out the component of gravity perpendicular to the hill's surface. The component of gravity that remains to move the ball is parallel to the surface:
where θ is the angle of inclination, and the component of FS perpendicular to gravity is
This force FP, parallel to the ground, is greatest when θ is 45 degrees.
Let Δh be the uniform interval of altitude between contours on the contour map, and let Δx be the distance between two contours. Then
so that
However, on a contour map, the gradient is inversely proportional to Δx, which is not similar to force FP: altitude on a contour map is not exactly a two-dimensional potential field. The magnitudes of forces are different, but the directions of the forces are the same on a contour map as well as on the hilly region of the Earth's surface represented by the contour map.
Read more about this topic: Scalar Potential
Famous quotes containing the words altitude, potential and/or energy:
“On a level plain, simple mounds look like hills; and the insipid flatness of our present bourgeoisie is to be measured by the altitude of its “great intellects.””
—Karl Marx (1818–1883)
“Much of what contrives to create critical moments in parenting stems from a fundamental misunderstanding as to what the child is capable of at any given age. If a parent misjudges a child’s limitations as well as his own abilities, the potential exists for unreasonable expectations, frustration, disappointment and an unrealistic belief that what the child really needs is to be punished.”
—Lawrence Balter (20th century)
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—C.S. (Clive Staples)