How does Newton’s second law relate to inertia?
The more inertia (momentum) an object has, the larger an impulse it will take to bring it to rest, or accelerate it a given amount and that impulse can be a large force acting for a small time, or a small force acting for a large time.
Let’s begin by defining inertia. “Inert” means static or unchanging. So in physics, inertia means the tendency of a body to continue in its state of uniform rest or linear motion unless that state is changed by a resultant force. describes this tendency. So your question could be rephrased as “How does relate to Newton’s first law?”
The inertia of an object is proportional to both its mass and its velocity. So our quantitative proxy for an object’s inertia is the product of both of these quantities. We call that product momentum:
##vec p = m vec v##
Note that momentum (and therefore inertia) depends equally on how much stuff there is and how fast it is moving.
Newton’s second law is
##vec F_{text{net}}= frac{dvec p}{dt} = (d(mvec v))/dt = m frac{partial vec v}{partial t} + vec v frac{partial m}{partial t}##
Where, if you’ve taken some calculus, the third equal sign is an application of the product rule. Now, noting that ##vec a = partial vec v## / ##partial t##, this becomes
##vec F_text{net} = m vec a + vec v frac{partial m}{partial t}##,
and in the case of constant mass, the second term on the right (which is called “thrust”) will be zero, and Newton’s second law reduces to the familiar
##vec F_text{net} = m vec a ##
The resultant (or net) force acting on a body is equal to the rate of change of momentum. So we can rearrange the second equation in this answer as
##d(mvec v) = vec F_text{net} dt##, or
##Delta vec p =int vec F_text{net} dt##
To produce a given change of inertia (momentum) in an object will require a resultant force acting for a period of time. The same change in inertia can be produced with a larger res. force acting for a shorter time or with a small res. force acting for a longer time. This integrated product of force times time is called impulse, and the final equation above is called the momentum-impulse theorem.
The more inertia (momentum) an object has, the larger an impulse it will take to bring it to rest, or accelerate it a given amount. That impulse can be a large force acting for a small time, or a small force acting for a large time.
