If an object under a constant force , say gravity. Collides with another object with no external force acting on it. Will the momentum of the system right before and after the collision be the same if the collision is instantaneous?
Assume that the collision is vertical.
[Physics] Conservation of Momentum with external force
collisionconservation-lawsmomentumnewtonian-mechanics
Best Answer
The Impulse-momentum theorem is $$\Delta \vec{p}=\int \vec{F}\cdot d\vec{t},$$ where the quantity on the right is the impulse, and $\vec{F}$ is an external force (or a sum of external forces). So if you want to know when momentum is conserved, $\Delta \vec{p}=0$, you want to know when the impulse will vanish. This can happen in two different generic ways
In your situation $\vec{F}\neq 0$, so the system is not isolated. You're saying the collision is "instantaneous", which to me means $\Delta t \simeq 0$, so that integral vanishes. Since is vanishes, $$\Delta \vec{p}=\vec{p}_f-\vec{p}_i=0\rightarrow \vec{p}_i=\vec{p}_f,$$ i.e. the momentum right before the collision is equal to the momentum right after.
We need to be a little careful though, since there is an external force on this system. So if your "initial" state is a finite time period before the collision, or if your "final" state a finite time period after, momentum will not be conserved.