Confused about Impulse Encountered a problem that involves impulse while studying for my exam and I'm not sure how to even approach it. I know that momentum is conserved, but I'm not sure how to relate that to avg force. Maybe someone can help point me in the right direction? I know that it's in quadrant III, through intuition, but I can't come up with a provable explanation
Relevant equation: $J=F_{avg}\Delta T$

 A: This post has some information about impulse that you might find useful. 
Homework Question involving Momentum
You will not find conservation of momentum useful here. True, the total momentum of object + wall is unchanged by the collision. But the momentum of the object does change. 
Since $\Delta P = J = F_{avg} \Delta t$, the direction of $F_{avg}$ and $\Delta P$ must be the same. 
A: The total impulse is the change in momentum (note that this is a vector equation):
$$ \vec{I} = \vec{p}_{final} - \vec{p}_{initial} $$
You know the momentum before and after the collision so you can calculate the total impulse, both magnitude and direction. Impulse if force times time, so the direction of the force will be the same as the direction of the impulse.
A: Alternatively, and qualitatively, think about the components of velocity (in the x y directions)  have changed.  Along the x axis, velocity has reduced, so the re has been a force in the -x  direction.  In the y axis, velocity has changed sign, so there must have been a force in the -y direction.  Hence the total force is down and to the left, ie quadrant III.
A: Let's start from the beginning-
Momentum is the product of mass and velocity of an object. Often written as
$$p = mv\tag1$$
(so more the mass or more the velocity, more the momentum. Think - you can give enough momentum to a ping-pong ball to equal momentum of a football moving at a certain velocity)
Impulse is the change in momentum of an object, often as a result of collision where the impact time is very small. An example is a bat hitting a ball
Thus Impulse, often denoted by
$$J = \Delta (mv)\tag2$$
Now we also know that when a bat hits a ball for that microsecond, it delivers a force on to the ball. Force in turns accelerates the ball which therefore implies that there is a change in velocity of the ball. We just saw in Equation 2 that a change in velocity provides impulse. So there should be a relationship between Force and impulse as well.
Take Equation 1 and differentiate both sides. What you will find is that
$$dp/dt = m dv/dt$$ or
$$dp/dt = ma$$ (remember a = dv/dt)
but we know that $F = ma$ therefore,
$$dp/dt = F$$ or
$$dp = F dt$$
integrate both sides and what you get is
$\delta (p) = F \delta (t)$$
or $\delta (mv) = F \delta (t)$
thus change in momentum is also equal to the force impressed multiplied by the time the force has been impressed. or-
$$\text{Impulse}\ J = \Delta (mv) = F \times \Delta (t)$$
So if the ball is hit with a force of 10 N for 0.1 second, the impulse transmitted is
$$J = 10 \mathrm N \times 0.1 \mathrm s = 1 \mathrm{Ns}$$
But remember, Newtons 3rd law say that if an object A impresses a force F on another object B, the other object B impresses the same force on the object A.
Since impulse is a direct function of force, this law applies to impulse as well. Impulse delivered by the bat to the ball is 1 Ns, the ball also delivered the same impulse to the bat in magnitude but opposite in direction.
Watch this video from The Science Cube made by me to simplify the concept further
What is Impulse

Another numerical that is somewhat like the problem you've presented. Watch the video answer I created to explain the same
Impulse on the ball| Force on the floor
