What is the derivative of kinetic energy with respect to velocity?

Answer 1

It's the linear momentum #p=mv#.

The kinetic energy of a particle is defined as #K=1/2 mv^2#.

It's derivative with respect to the the velocity #v# is:

#(dK)/(dv)=d/(dv)[1/2 mv^2]#

Since the mass #m# does not depend on the velocity and the factor #1/2# is constant, the linear property of the derivative gives us:

#d/(dv)[1/2 mv^2]=1/2 m d/(dv) [v^2]#

Knowing the derivative of a power function #d/(dx)[x^n]=n x^(n-1)# gives us the result:

#(dK)/(dv)=1/2 m 2 v=mv=p#

This answer is valid if we consider the classical case. Taking into account relativistic effects gives us the same result, but the derivation is more complicated.

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Answer 2

Isaiah Allen

The derivative of kinetic energy with respect to velocity is momentum.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.