Newton’s Second Law in Terms of Momentum

When Newton’s second law is expressed in terms of momentum, it can be used for solving problems where mass varies, since Δ𝐩=Δ(𝑚𝐯)Δp=Δ(�v) . In the more traditional form of the law that you are used to working with, mass is assumed to be constant. In fact, this traditional form is a special case of the law, where mass is constant. 𝐅net=𝑚𝐚Fnet=�a is actually derived from the equation:

𝐅net=Δ𝐩Δ𝑡�net=Δ�Δ�

For the sake of understanding the relationship between Newton’s second law in its two forms, let’s recreate the derivation of 𝐅net=𝑚𝐚Fnet=�a from

𝐅net=Δ𝐩Δ𝑡�net=Δ�Δ�

by substituting the definitions of acceleration and momentum.

The change in momentum Δ𝐩Δp is given by

Δ𝐩=Δ(𝑚𝐯).Δp=Δ(�v).

If the mass of the system is constant, then

Δ(𝑚𝐯)=𝑚Δ𝐯.Δ(�v)=�Δv.

By substituting 𝑚Δ𝐯�Δv for Δ𝐩Δp, Newton’s second law of motion becomes

𝐅net=Δ𝐩Δ𝑡=𝑚Δ𝐯Δ𝑡�net=ΔpΔ�=�ΔvΔ�

for a constant mass.

Because

Δ𝐯Δ𝑡=𝐚,ΔvΔ�=a,

we can substitute to get the familiar equation

𝐅net=𝑚𝐚Fnet=�a

when the mass of the system is constant.

TIPS FOR SUCCESS

We just showed how 𝐅net=𝑚𝐚Fnet=�a applies only when the mass of the system is constant. An example of when this formula would not apply would be a moving rocket that burns enough fuel to significantly change the mass of the rocket. In this case, you can use Newton’s second law expressed in terms of momentum to account for the changing mass without having to know anything about the interaction force by the fuel on the rocket.