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Name: Danielle
Status: student
Grade: 9-12
Location: CA
Country: N/A
Date: 12/13/2005

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.

Vince Calder

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