The goal is to study “forces”.

Can we quantify “forces”?

Consider a wooden block on a smooth horizontal surface. Suppose the wooden block is initially moving from west to east with a velocity ${ \vec{v} . }$

i) Note that if no “force” acts on it, it will continue to move with this velocity indefinitely.

ii) Suppose a constant “force” acts on it, pushing it towards north, for a time interval ${ t _1 }$ to ${ t _2 . }$ Then in the process, the velocity will tend to change in the direction of “force” applied.
We can guess

\[{ \text{Guess:} \quad \Delta \vec{v} = \vec{c} \Delta t }\]

for each small time interval ${ \Delta t , }$ where ${ \vec{c} }$ is a vector in the direction of the “force” applied and depending only on the “force”.

iii) Suppose the mass of the block is changed to ${ m _2 » m _1 . }$ Suppose the same constant “force” as in ii) acts on it, pushing it towards north, for the same time interval ${ t _1 }$ to ${ t _2 . }$
Then the change in velocity is smaller than in ii).
Hence to account for the effect of mass, we can guess

\[{ \text{Guess:} \quad \Delta \vec{v} = \frac{\vec{c}}{m} \Delta t }\]

for each small time interval ${ \Delta t , }$ where ${ \vec{c} }$ is a vector in the direction of the “force” applied and depending only on the “force”.

This suggests the equation almost defining force,

\[{ \vec{F} \propto \vec{c} = m \frac{d \vec{v}}{dt } . }\]

We quantify force and choose the units of force such that

\[{ \vec{F} = m \frac{d \vec{v}}{dt} . }\]

For example, a “${ 6 }$ ${ \text{kg} \, \text{m} / \text{s} ^2 }$ ${ \, \hat{x} }$” force is defined to be that force which changes, in a small time interval ${ \Delta t }$, the momentum of a particle by ${6 \Delta t }$ ${ \text{kg} \, \text{m} / \text{s} }$ ${ \, \hat{x} . }$

Forces are defined by the change in momenta they cause in matter.