Q.One of the effects of a force is to change the momentum. Define the quantity related to this and explain it for a variable force. Usually when do we define it instead of using the force?

[!TLDR] The quantity is IMPULSE, $\vec{J}=\int \vec{F}\,dt=\Delta\vec{p}$; it is defined and used instead of force whenever the force acts for such a brief, hard-to-measure time that only the resulting change in momentum can practically be measured. [!ANSWER] Imp …

The quantity related to a force's effect of changing momentum is called IMPULSE, $\vec{J}$. For a constant force acting over a measurable time t, $\vec{J}=\vec{F}t=\Delta\vec{p}$; for a VARIABLE force (one whose value changes with time during the interaction), impulse must instead be computed by integration, $\vec{J}=\int \vec{F}\,dt=\int d\vec{p}$, i.e. the area under the force-versus-time (F-t) graph. This is usually defined and used INSTEAD of directly quoting the force whenever the force acts for an extremely short, difficult-to-measure interval of time -- situations like a bat striking a ball, a foot kicking a football, a hammer striking a nail, or a ball bouncing off a hard floor, where the peak force during contact is large but the contact TIME is too brief to record accurately on its own, while the resulting CHANGE IN MOMENTUM of the object is a perfectly well-defin …

Define impulse from the second law's integrated form, and give its variable-force (integral) generalisation, explain …

Defining impulse only for constant forces and forgetting the integral $\int F\,dt$ need …