What is the integral of the momentum function?

Question

Ezra Adams · Accepted Answer

The momentum formula is typically given by #p = mv#, where #p# is momentum, #m# is mass, and #v# is velocity. One must first decide whether one wishes to integrate with respect to velocity or with respect to mass. If one integrates the function with respect to velocity (and thus treats momentum as a function of velocity), one receives:

#int p(v)dv = int mv dv#.

If we assume that mass is constant, then we can factor it out:

#m int v dv#

Then, by using the power rule with respect to integrals:

#m int v dv = m 1/2v^2 = 1/2mv^2#

Note that this is equivalent to the formula for kinetic energy.

Charles Arocha · Answer

undefined The integral of the momentum function, denoted as ∫p(t) dt, represents the change in momentum over a specified time interval. It is calculated by integrating the momentum function with respect to time over that interval. Mathematically, if the momentum function is given as p(t), where t represents time, then the integral of p(t) dt from time t₁ to t₂ (denoted as ∫p(t) dt from t₁ to t₂) can be found by evaluating the antiderivative of p(t) with respect to t over the interval [t₁, t₂]. The result represents the total change in momentum during that time interval.