So you seem to be having trouble understanding Newton’s laws. I’ll try to address that first, and then later answer your questions directly. It’s inevitable that I will repeat something you are already familiar with, but I think you need some refreshing.
Just to cover all bases: Newton's first law describes how a rolling wheel will keep on rolling and rolling. It will never stop unless it is stopped by an outside influence (like friction, or a tree). A bit after it runs into the tree, it'll become a stationary wheel. Since it's at rest, it'll keep being at rest. Unless something (like a foot) applies a force upon it and changes it’s state.
Now merely apply the fancy wording:
First law: The velocity of a body remains constant unless the body is acted upon by an external force.
In the case of the rolling wheel, its velocity, which is non-zero (as it is in the state of motion), remains constant until it is acted upon by an external force (the force applied on the wheel by tree). After this event, its velocity will be reduced to zero and the wheel will be in a state of rest. It will not leave this state of rest unless it is acted upon by an external force.
Let's take the situation you described: "When you have a problem where one object collides with another after being pushed (given the force, mass of both objects), how do you find the acceleration of the first object, and the force exerted on the second?"
And let's also make it a bit less abstract (though the values we'll use will be unrealistic):
You have a block of cheese and a knife. You cut the cheese with the knife. The cheese requires a force of 10N to cut. So the knife was initially still, then it moved in order to cut the cheese. In other words it accelerated. But what caused this acceleration? The answer is a force. Ignoring the situation where the force is not big enough to do anything useful; If you apply a force to something, it will change what it was doing (Newton’s First Law). It’s velocity will change – again in other words it will accelerate
.
Of course, you can have more than one force acting on an object. For this reason we refer to all the forces that act on an object as the
net force. In our simple situations, we usually only consider one force that has acted, so that single force will be our net force. This is an important distinction to make.
So, let’s summarise: we know that a net force causes acceleration. We also know that this effectively changes the
inertia of an object. So net forces are proportional to inertia and acceleration. But we want hard numbers – what’s the measure of inertia?
Mass. The more mass an object has the more inertia it has. In other words, an object with greater mass will be harder to change its state of motion.
So there we have it:
.
This should fit quite intuitively with your idea of weight. If you have a light person and a heavy person and you pushed them both with an equal amount of force, the heavier person wouldn’t move as far. If you wanted to accelerate the heavy person with the same acceleration that the skinny person had, you would have to push harder (apply a greater force).
So, back to our cheese and knife. We need to apply a force of 10N and in our simple situation, we use the knife to apply the force. Since this is the only force we’re talking about, we apply a
net force of 10N.
Newton’s Third Law, is probably the hardest to understand (but easiest to remember)
“For every action there is an equal and opposite reaction”
Push your knuckles together, as hard as you can. The amount they discolour will be the same, because they both experienced equal and opposite forces.
Note that it doesn’t matter which you consider to be an action and which you consider to be a reaction. Just don’t switch around half way while working out. You don’t talk about the action and then suddenly switch to talk about the reaction. I think this is the point where you were getting confused was where to apply Newton’s third law.
You also need to remember that the Newtonian pair must act on different objects. Let’s use the example of a person (let’s call him Howard) standing on the Earth. So, due to gravity, Howard is applying a force on the Earth (this will be force 1). The Earth is applying a force on Howard (this will be force 2). Force 1, which is Howard on Earth, cannot act on Howard. Force 2, cannot act on the Earth. It is Earth on Howard. If they acted on the same object, the forces would cancel each other out and we would live in a world where no force could ever be applied.
(I don’t know if this paragraph is 100% correct – someone please confirm this). So, we want to know acceleration of the knife. So we had:
. This was the force of the knife on the cheese. The force of the cheese on the knife would be equal AND opposite. That would be -10N. We’re going to ignore that fact though, we’re only talking about the action here (though the complete
interaction is comprised of the knife acting on the cheese and the cheese acting on the knife).
Let’s make up some values for the masses now:
We'll now consider the force of knife on the cheese, and calculate it's acceleration.
(yeah, ridiculous behaviour for whoever was using the knife)
Why is it accelerating in a positive direction? Because we defined the force of the knife on the cheese to be positive.
Let's take a look at the
reaction:
So, the acceleration due to the reaction is negative because we defined the action to be +ve.
You could easily switch around the positive and negatives if you wish.
The force exerted on the cheese was 10N (0.500*20), while the force exerted on the knife was -10N (1*-10)
I know I didn’t address your entire question, but hopefully this cleared things up.
You don’t have to give directions if it only asks for the magnitude. You don’t always have to mark F and a as vectors with an arrow if you don’t want to – just don’t forget they are actually vectors.
Technically you should be always providing directions, if asked for acceleration. VCAA exams will get you in the bad habit of leaving it out (very rarely will they want directions).