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Completely Inelastic Collisions Recall earlier that collisions with energy loss are called inelastic. Objects still bounce off of each other in these collisions, but their velocities will be a little less compared to those found in the last section. How much less depends on how much energy is lost during the collision. In completely inelastic collisions, the objects actually stick together after the collision, so they become one object with one velocity. Therefore, the momentum conservation equation reduces to,  Because both objects stick together into one, the masses are added multiplied by their velocity after the collision. There is no need for a number after that velocity since both objects share the same value, so it is just called V prime. Notice how this type of collision does not have the same relative velocity property found in elastic collisions. There is no relative velocity after the collision because both objects are traveling together. It turns out that this is not the only difference as we will see in the next example. Suppose a car with a mass of 1000 kg traveling at 20 m/s runs into another car with the same mass at rest on an icy road and the collision is completely inelastic. Find the velocity of both cars after the collision. Because the second car is at rest, the momentum equation becomes, Which becomes, V' is 10 m/s. So, both cars would be traveling at 10 m/s right after the collision. Of course they would not keep that speed up due to friction. Because of the icy road, they would have that speed right after impact.  Notice that the final speed was exactly half the original speed. This is a result of both masses being the same. It turns out that if the moving object has a greater mass than the object at rest, the resulting velocity will be greater than half the original. On the contrary, if the moving object has a smaller mass than the object at rest, the resulting velocity will be less than half the original. The velocity after the collision is only half the original when both masses are equal.  What about energy conservation with the last example? If we plug in the numbers for the initial energy we will have, The final energy is found by plugging in 2000 for the mass, since both objects are as one, and 10 m/s for the speed. The initial and final energies are not equal! The reason: energy is lost in completely inelastic collisions. Where does it go? The energy is lost in the impact because both objects will become deformed since they stick together. This deformation results in a loss of energy. Think of it this way. In an elastic collision, energy is lost but both objects still bounce off of one another. But, the energy is still lost through the impact because both objects can still sustain some damage. In the case of a completely inealastic collision, even more damage is sustained because they become entangled. Does this mean energy conservation is violated? No! Remember, as long as all of the energy is accounted for, no law is broken. What happens if both objects are moving? What if a car traveling at a certain speed hits another car head on with a smaller mass and is also moving with a certain speed. Which direction would the two cars head after the collision? Would the two cars always continue in the same direction that the larger mass was originally heading? Not quite. It all depends on who has the larger momentum. The larger momentum would win, and the two objects would continue traveling in that object's original direction after the collision. To finish up this section, let's consider a two-dimensional collision. Say a 20.0-kg ball is traveling in the positive x-direction at 15.0 m/s. Ball two with a mass of 10.0-kg is traveling in the positive y-direction at 60.0 m/s. If they collide, find the speed and direction of the clump.   This problem is dealing with x and y-components, so we need to work with each component separately to find V prime's components.  Momentum in the x-direction Only object one is moving in the x-direction, so the momentum equation becomes, Vx' equals 10 m/s. Momentum in the y-direction Only object two is moving in the y-direction, so the momentum equation becomes, Vy' equals 20 m/s. To find V', we need to use the pythagorean formula, To find the angle, we need to use the inverse tangent function, Therefore, the clump has a velocity of 22 m/s 63 degress above the horizontal. Notice how most of the motion of the two balls after the collision was in the y-direction. This was due to the fact that ball two's initial momentum, which was in the y-direction, was larger than ball one's initial momentum. Therefore, ball two had more of an impact on the final velocity of both of them together. Had ball one been the object with the greater initial momentum, the angle would have been more towards the x-direction. In fact, if both initial momenta were equal, the angle after the collision would have been right down the middle of the two directions, which would be 45 degrees. Newton's Apparatus Most people are familiar with Newton's apparatus, which consist of five steel balls that hang in a row.  The popular thing to do is to pull one of the balls back and watch as the ball on the other end pops up. The two end balls take turns popping up until they lose all of their energy and stop.   This apparatus is a very good illustration of momentum and energy conservation at work. When an end ball is pulled back, it slams into the first ball and since they are all the same mass, it gives all of it's energy and momentum to the first ball. However, it can't go anywhere because it immediately hits the second ball, giving up all of its energy and momentum. This happens until the last ball is hit, and since it has nothing in its path, it flies off with the same energy and momentum as the original end ball. In reality, the last ball's energy and momentum is a little less due to the loss of energy from each collision. What happens when two balls are pulled back?  Would it be possible for the end ball on the right to fly off with twice the speed as the original two, and therefore go up to twice the height?  No matter how many times you might try, just one ball will never fly off with twice the speed. Even though momentum would be conserved, energy would not. This is the same situation covered in the last section. Say the mass of each ball is 1.0-kg, and the two balls hit the next ball at a speed of 5 m/s. The initial energy would be, However, if the end ball had a speed of 10 m/s, the final energy would be, The final energy would be twice the original, which cannot happen. What does happen? Two balls fly off.  What happens if three balls are pulled back?  You know that the two balls that are left cannot fly off with a speed greater than the original three because that would violate energy conservation. What is the only possibility that would conserve both momentum and energy? Three balls have to be moving after the collision and two must be at rest. Is that possible with this situation? It is possible if the third ball in the original group joins the two balls that are at rest before the collision so that three balls are flying off after the collision. In fact, that is exactly what happens!  It should seem clear by now that there is only one possibility for each starting position, because both momentum and energy have to be conserved at the same time! As a result, there needs to be perfect symmetry. For example, if three balls are moving and two are at rest before the collision, three balls must be moving and two must be at rest after the collision. This fact will now make it easy to predict any situation. Say three balls are moving to the right and two balls are moving to the left before the collision. This means three balls must be moving to the left and two balls must be moving to the right after the collision.   The big key to all of these collisions is summed up with this apparatus. Momentum and energy have to be conserved at the same time, which limits the collisions to having one possible outcome. Otherwise, there would be many possible outcomes which would make for a very unpredictable Universe. In fact, if one of the conservation laws were found to be broken in a collision, the consequences would be devastating to our present knowledge of the world as we know it. Because both laws are obeyed, every collision from cars to air molecules is able to be predicted, which keeps order in the Universe. |