Impulse help emergency.(3 problems)

[crimsonfears]

Impulse help! emergency.(3 problems)

Alright, so I have these problems due tomorrow, and I was out the other day so I do not understand it. If someone can tell me how to set each up, then I can probably work it out.
Equations
impulse= Force*$$\Delta$$Time
Force*$$\Delta$$Time=mass*$$\Delta$$velocity
momentum=mass*velocity
Force=mass* acceleration

(unsure if the last two equations are necessary for problems)

The first problem:
Two children, totaling 200 kg, are traveling at 10 m/sec in a 100-kg bumper car during an amusement park ride. They deliberately collide with an empty second car, mass 100 kg, which is at rest. Afterwards, the car with the two children moves off at a speed of 4.0 m/sec. What is the final velocity of the empty car?

The Second problem:
James, a 65-kg skin diver, shoots a 2.0-kg spear with a speed of 15 m/sec at a fish which is darting past him. How fast does James recoil when the spear is initially released?

The Third problem:
On a hot summer day, Jack and Leon are fishing in their boat, when they decide to jump into the water to cool off. Jack, 45-kg, jumps off the front of the boat with a speed of 2 m/sec. While at the exact same moment, Leon, 90-kg, jumps out of the back of the boat at a speed of 4 m/sec. If the boat has a mass of 100 kg and was at rest prior to the two boys jumping off, what will be its velocity just after both boys have abandoned ship?

 

[crimsonfears]

Oh, and for the first problem, I got some obscure answer. Here's my work.
(w/ kids)
m=200kg+100kg
Vi=4.0m/s
Vf=10m/s
(empty car.)
vi=0 m/s
v/f=?
m=100kg

m$$\Delta$$v=300*(4-10)=-1800kgm/s
-1800=100(x-0)
/100
Vf= -800 m/s

but that seems way off.

 

[hotcommodity]

You're right about your answer being way off, but it's good that you recognize that. Try to forget about the numerical values of the masses and speeds involved in the problem for a minute. I think I see where you made and error, and it has to do with the way you set up the momentum equation. Can you show me how to set up the equation using variables only?

 

[crimsonfears]

uhh, i know this much lol..
p=mv

 

[hotcommodity]

uhh, i know this much lol..
p=mv

Well, that's a start. Think about this, we have one cart running into another cart. I want you to picture both carts in your mind, and separate them from everything else around them. When we do this, we create a "system." And the system only consists of those to carts.

Now we have an event taking place in that very system, the event being the one cart smashing into the other. If no net external average force acts on our system, we can say that the momentum of the system is conserved. Above, you defined momentum as p = mv. That equation however can only be applied to one object. We want to consider the momentum of our entire system, that is to say we want to consider the momentum of each cart, both before and after the collision.

The question is, what equation should we use to find the final velocity of the empty cart? You were speaking of the "impulse" concept above, so you should know that the impulse is mathematically defined as the average force times the time interval that the force occurs, that is:

$$I = F_{ave}t$$

The impulse is also defined as the change in an objects momentum:

$$I = \Delta p$$

Now we know that $$F_{ave}t = \Delta p$$, and therefore, when no net average force acts on an object, or in this case a system, we have:

$$p_{i} = p_{f}$$

If this is the case, we say that the momentum of the object, or the system, is conserved. As I said above, we want to consider the TOTAL momentum of the system before and after the collision. Does any net force act on our system? The answer is no. When we consider Newton's Second Law, we see that the normal force on each cart cancels with the weight force of each cart. Now we know that the linear momentum of our system is conserved, so we can say that the final momentum of the system is equal to the initial momentum of the system:

$$P_{total, f} = P_{total, i}$$

$$M_{cart 1}*v_{cart 1, f} + M_{cart 2}*v_{cart 2, f} = M_{cart 1}*v_{cart 1, i} + M_{cart 2}*v_{cart 2, i}$$

Now all you have to do is solve for the final velocity of cart 2, and plug in the values that you were given. I know this seems like a long explanation, but the more you know about what's going on, the easier it will be to identify the main physics concept involved in each problem. Let me know what you get :)