The Derivative of Kinetic Energy

The formula for kinetic energy is

KE = ½mv²

What is the derivative of kinetic energy?For a body that starts from rest and travels with constant acceleration, we have the following formula

v ² = 2ad

Multiplying by ½m

½mv ² = mad

A graph of this equation is a straight line. The slope of the line is ma.The slope of a tangent line on a graph of a function is the derivative of the function.The derivative of kinetic energy, with respect to distance, is mass times acceleration.

Newton's Gravity Theory

The formula for force is

F = ma

The formula for the force of gravity is

F_g =	GMm
	r ²

G is a constant, M and m are the masses of two bodies, r is the distance between their centers.The force of gravity between the earth and a body on its surface is the weight of the body.Isaac Newton came up with these two formulas in 1687.

My Gravity Theory

A body weighs less when it is moving.The rest weight of a body is how much it weighs when it isn’t moving.The formula for rest weight is

RW =	GMm
	r

Weight is equal to rest weight minus kinetic energy.

Falling Bodies

What is the acceleration of a falling body?

Newton's Way

The acceleration of a falling body is represented by g.The force acting on a falling body is the force of gravity. For a falling body, we have

F = F_g

mg =	GMm
	r ²

g =	GM
	r ²

My Way

The formula for weight is

W = RW − KE

The rest weight of a falling body is increasing.The kinetic energy of a falling body is increasing.The weight of a falling body stays the same.Differentiating this equation

DW = DRW − DKE

For a falling body, the derivative of weight is zero.

The derivative of rest weight is

DRW = −	GMm
	r ²

Substituting

0 = −	GMm	− ma
	r ²

a = −	GMm
	r ²

The positive direction is up. The acceleration of a falling body is downward, so the acceleration is negative.

Wrong

The formula for force is

F = ma

The derivative of kinetic energy is

DKE = ma

One of these formulas is wrong.

Newton never heard of energy. Newton didn’t know what he was doing.

Springs

Gravitational potential energy is the integral of the force of gravity. The formula for gravitational potential energy is

RW =	GMm
	r

Hermann Helmholtz came up with this formula in 1847.

Wrong Formulas

Helmholtz's Way

RW =	GMm
	r

My Way

Backwards

Negative energy is stupid.

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