Ok, I got an impulse problem here

[alberto]

Ok, here the problem my teacher gave:

The goal is to close an open door in the best possible manner. You have two objects, a bouncy ball and a big chunk of sticky mud. You decide to throw one of these objects at the door to close it. The bouncy ball will bounce off the door with the same speed as it struck the door. The mud will stick to the door. Which of these two scenarios will result in the door closing with a larger speed? Use the concept of impulse in your answer.

So at first I was like "the ball", since with Impulse being the change in momentum = force x time, with the ball hitting the door and bouncing off really fast, time would have to go down in the equation, thus, force would have to go UP to compensate. But then I rationalized that if the ball is coming back off the door in the opposite direction at the same speed, it’s momentum was pretty much conserved and therefore not much was transferred to the door to shut. The mud on the other hand isn’t bouncing away, thus all of its momentum will strike the door and continue to push on the door in the same direction that the door needs to go in order to shut.

so there is definitely something wrong with one of these lines of thought...help! and thanks!

 

[FZ+]

Yep, here's your problem...

But then I rationalized that if the ball is coming back off the door in the opposite direction at the same speed, it’s momentum was pretty much conserved and therefore not much was transferred to the door to shut.

Now, remember that momentum is a VECTOR quantity. While the speed is the same, the velocity and therefore momentum of the ball is entirely different. Not only did the ball exert an impulse on the door to stop moving, but it also exerted an impulse to give the ball an momentum in the opposite direction.

 

[alberto]

So in terms of my 1st answer, the ball would then be applying greater force than the mud since its time in contact with the door would be less than the mud, right? (thanks for the quick reply!)

 

[quantumdude]

No, that's not what he's saying. When he alludes to the vector character of momentum, he is getting at the fact that the sign is important. When the mud hits the door, its momentum changes from p to approximately zero (I say approximately because the mud moves with the door after impact). On the other hand, when the ball hits the door its momentum changes from p to -p.

Can you take it from there?

 

[FZ+]

Also, the time thing is a red herring. While the force exerted by the ball would be greater, the time taken to exert the force would be smaller. These cancel each other out.

Hitting someone with kilo of steel would hurt more than a kilo of bean bags, but it would knock them back by the same amount.

 

[alberto]

whoa whoa whoa...
I understand regarding the kilo of steel vs. kilo of bean bags, and I get how the time of the impulse is less so the effect of the force is not experianced to the same extent. So is the answer that because the momentum of the mud is in the same direction (with an additive effect) as the way the door is going to close, that this would result in the door closing with greater speed? My line of thinking is that the door's mass + the mass of the mud (since it sticks) means that the door would have to go from an initial velocity of 0 (while it's open) to some final velocity when the mud hits (when it's shutting)that would be greater than the ball hitting since the ball doesn't stick...I think I'm confusing myself (again)...

thanks again for the replies!

 

[FZ+]

Ok... let's try this again.

conservation of momentum states that mu = mv + Mw

m = mass of ball, u = initial velocity of ball, v = final velocity of ball, M = mass of door, w = final velocity of door. All velocities are positive in the same direction.

now, with mud...

mu = (m + M) * w
or, Mw = mu - mw

Since the ball is now the same velocity as the door.

with the other ball...

mu = Mw + m * (-u)
since the ball is travel at the same velocity in the opposite direction to what it was before.

Rearranging...
Mw = 2 * mu

Obviously, Mw(momentum of door) is now greater. Do you see?

 

[alberto]

That explanation was perfect! I get it! thank you soooooo much! When I sat down and did the algebra, it made sense. Thanks again, good karma to you!