Momentum and Kinetic Energy
Why is momentum the derivative of kinetic energy?
This relation follows from the definitions: kinetic energy, KE, is defined
as: KE = 1/2 x M x V^2 and momentum, P, is defined as: M x V where M is the
mass and V is the velocity, so: d(KE)/dV = 1/2 x d(M x V^2) and if the
mass is a constant: d(KE)/dV = 1/2 x M x d(V^2/dV) = 1/2 x M x 2 x V = M x
V = P. There is one further refinement that needs to be taken into
consideration, but it does not change the overall result. Velocity is a
vector quantity (that is, it has both magnitude and direction so strictly
speaking V = (Vx, Vy, Vz) and V^2 is, strictly speaking, the dot product of
V and itself, V*V = Vx^2 + Vy^2 + Vz^2. Momentum on the other hand is a
vector, and so has both magnitude and direction so strictly speaking it is
the magnitude of P: |P| = [M x (V*V)^1/2]. In introductory presentations of mechanics this
refinement is usually ignored.
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Update: June 2012