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Momentum

I will start with some hisotrical background and then relate this background to the collision lab we did. When we discussed Newton’s Laws we already talked about how we are trying to find a cause for the motion. However in with forces we only found out what causes motion to change. The question when does motion stop or why does it keep going seems simpler to answer then figuring out what causes object to change their motion.

After Aristotle, who talked about the natural order of things being the cause for motion, it was a Persian philosopher who looked further into motion.  In the 11th century Ibn Sina studied the motion of projectiles and assumed that moving objects contain a quality that is related to their heaviness and given to them by the person or thing that made the object move in the first place. He called this quality “mayl” and for him it was the things that “helped carry the motion”. It was a quality of the object itself.

Three hundred years later this idea was changed a little with Jean Buridan (who only lived to be 30 years old) talking about motion as being a property that is given to the object. Much like a coat of paint that is put on a ball. This paint will fade but only if wind or sun or rain will make it erode. He called this property “impetus”. Impetus is given to an object when it is made to move. Air resistance and wind ca decrease the impetus and gravity can increase the impetus. Both of these are outside forces. Without outside forces the impetus would continue forever. Heaviness of the object can also change its impetus.

 

In class we did a lab in which we had a moving cart collide with a stationary cart. After the collission the carts stuck and moved off together. This lab allowed us to look into how the four variables (two masses and two velocities) are related to one another and as a conclusion to the lab we found one relation - one equation - that links those four variables. While the first line on the left is the relationship we foudn in the lab, the last line is the same relationship with the variables resorted. Thye are resorted so that on the left side we describe what is moving before the collisiosn (cart 1) and on the right side what is moving after the collision (cart 1 and 2 together) .

If we look closely, we find that the quantities on either side of the eqaul sign are of the same mke-up - they contain mass and velcoity. We could say that this quantiy describes a mass in motion.

But not only that, this quantity also stays the same during a collision - the "mass in motion" before the collision is equal to the "mass in motion" (now two carts moving) after the collision. It is said that that the quantity is "conserved". It makes therefore sense to look at this quantiy of motion a little more closely realizing that it does not change in a collision. Somehow this must be important.

This relation of mass to motion (or velocity) is important because so far we have thought of motion as independent of mass. The combination of mass and velocity we now call momentum, the Latin term for movement. In the lab we did we figured out that momentum is a quantity that can be transferred form one object to the next if these two objects come in contact, that is, if they collide. When a big (moving) ball collides with a small ball at rest some or al of the motion gets transferred from the big to the small ball. This can also work the other way if a small ball collides with a big ball, but it is easiest to see when two of the same balls collide. In the lab we actually found a relationship between the momenta of the two objects and turned it into a mathematical equation, something that was far out of reach for geniuses like Newton or Galileo, because algebra was not a tool used to proof concepts back then.

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Going back to our lab and the conclusion we drew from our experiment, we can now say that the momentum of cart one before the collision is equal to the momentum of both carts after the collsion. (I could even the say the momentum of both carts before the collsion is equal to both carts after the collsion, since the second cart is initially not moving.) That there is such a quantity that does not change regardless what happens is very important and it is called Momentum Conservation. In physics very few quantities are conserved (mass and energy are the others we will discuss). Momentum conservation can be used to solve a variety of problems. Besides the "sticky" collision we discuseed in the first lab, there are "bouncy" collsions (second lab) as well as what i call "explosions" where the objects were at rest before but then explode and fly apart - the recoil of a gun would be a good example.

To solve problems involving momentum conservation is stright forward, as long as you include everything that moves initially (left side) and set it equal to everything that moves after the collision (right side). Consider this problem:

A 1000 kg car moving at 30 m/s collides head on with a 1500 kg car moving in the opposite direction with a speed of 10 m/s. The two cars crumple, stick together and move off together. How fast is the combined wreck moving just after the collision?

Solution: Initially both cars are moving, but in opposite directions, which means one of the velocities is negative. (Momentum is a vecotr.) So, on the left side we have both momenta (one negative) and one the right side we have both after the collision.

The cars are moving off together with a speed of 6 m/s.

We can think about this whole concept of motion and momentum from a different angle as well. Let’s think about how the term momentum is used in every day life. A soccer team can have a lot of momentum, or a stock on the stock market. There are momentum plays on the stock exchange. A football player can have a lot of momentum. What all of these have in common is that there motion will be hard to stop (or change direction in the stock example). In physics “momentum” can be interpreted in the same way. We could ask ourselves how hard it would be to stop an object. This depends on both mass and velocity of the object and therefore agrees what with what we discussed earlier.

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When we look at stopping of an object more carefully we will see that there are two different ways to stop an object. Both involve force. Without force an object cannot be stopped, because stopping is changing motion and changing motion can only be accomplished with a net force.  You can stop a cruise ship for instance, by applying a small force (friction) or by applying a big force (running it aground). The two forces are different (big and small) but something else involved in the process of stopping the ship is also different – the small force has to be applied over a long time, the big force can be applied over a short amount of time. There are many applications of this concepts from airbags in cars o how to handle a ball in lacrosse.

The force slows down the ship, therefore changing its velocity and because of that the momentum changes. This change in momentum, caused by a force, is called the impulse.

 

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