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Momentum Momentum is simply inertia in motion. Recall from chapter three that inertia is the object's resistance to changing states of motion, and is measured in terms of the object's mass. Motion deals with velocity, which means the equation of momentum depends on two things: mass and velocity. The equation is simply, Where p is the letter used to represent momentum. The equation means an object has momentum by virtue of having a mass and a velocity at the same time. Therefore, an object cannot have momentum when it is at rest. The unit of momentum is kgm/s, or a Newton-second. Impulse Recall that the amount of work on an object equals the change in kinetic energy expressed in the Work-Energy Theorem. There is a similar relationship that equals the change in momentum of an object. Instead of force times distance, it is force times the amount of time the force is exerted, which is called Impulse. Therefore, the impulse-momentum equation becomes,  This relationship is very similar to the work energy relationship, in that a force applied to an object will change its kinetic energy and momentum. However, the momentum relationship is not a square relationship like work and energy. Recall, in order to double the speed, you have to do quadruple the work, by either increasing the force by four, or the distance by four. (You can even double the force and double the distance to get the same effect). But, according to the impulse and momentum equation, in order to double the speed, you just have to double the force, because the speed in the momentum equation is not squared. This is easily verified by making the mass one kilogram, the time one second, the initial speed zero, and the final speed 5 m/s. The force required to accomplish this is found by, The force is 5 Newtons. Now, in order to double the speed to 10 m/s but still keeping the same time, the force will be 10 Newtons. This seems contradictory to the Work-Energy Theorem, but it is not. The reason has to do with time and distance. The force is multiplied by distance in the Work-Energy Theorem, and it is multiplied by time in the impulse momentum equation. Let's do an example to illustrate this point. We can use the same numbers above, so we know the impulse and momentum part. Now we need to determine the distance the object experienced the 5 Newton force. Using the kinematic equation, We need the acceleration, so we will use Newton's Second Law F = ma, Since the acceleration is 5 m/s^2, the distance can be found by, This gives a distance of 2.5 meters. So, if we plug that into the work energy equation, Both sides equal 12.5 Joules. So, we have shown that a 5 Newton force exerted for one second on a one kilogram object will cause it to go from zero to 5 m/s over a distance of 2.5 meters. Thus, both the impulse-momentum and work-energy equations are satisfied. Now, let's see what happens when the force is doubled. We know from the previous example, the speed will be 10 m/s if the force is doubled and is exerted for the same amount of time. That is the key here: the same amount of time. Since there is a different force, we need to find the new acceleration, Now that we know the acceleration is 10 m/s^s, we can find the distance the object experienced that force, This gives a distance of 5 meters. We can again show that the work done on the object does equal the change in kinetic energy, Both sides do indeed equal 50 Joules. Notice that this is four times the original energy, which shows that you needed four times the work to actually double the speed. The four times did not come from quadrupling the force, however. The force was doubled, but the distance was doubled as well, 2.5 meters to 5 meters. This results in quadrupling the work. So, we have now shown that a 10 Newton force exerted for one second on a one kilogram object will cause it to go from zero to 10 m/s over a distance of 5 meters. Even though the time the force was exerted remained the same, the distance the object experienced that force was twice as much. This is how both equations are satisfied at the same time. The diagram below sums up the entire example.  There are very practical applications to impulse and momentum. Consider the egg toss game. Two people start off five feet apart and toss an egg back and forth, each time backing up a step. How do people catch the egg? Do they catch it with a stiff hand with no give? No, because the egg would break. Instead, they catch the egg and move with the egg as they slow it down to a stop. The point of moving with the egg is to catch it with a smaller force so that it does not break. This is impulse and momentum in action. Consider the picture below with the concept equations. (Technically, the equations should have a negative MVo, but for the sake of this example I am leaving the negative off).  In the first catch, the hand was stiff and caught the egg with no give, thus resulting in a large force, because the time of impact was small. If time is small, the force has to increase because the product must equal the change in momentum. Therefore, if the time of impact is increased, like the second catch, the force will decrease to keep the product of the two still equal to the change in momentum. This concept can be related to so many examples in every day life. Below are just a few. Cars are built to crumble in a head on collision to increase the time of impact, which reduces the force of impact. Padding is used everywhere, such as shoes, gymnastic floors, carpets and carpets to help increase the time of impact. Padded shoes are especially useful when a person has to do a great deal of walking or running. Each hit of the shoe on the floor sends shock waves up through the body. The greater time of impact helps to reduce the wear and tear on the body over a long period of time. Boxing gloves are padded for obvious reasons. However, in addition to the padding boxers are taught to "roll" with the punches which means when a punch comes to the face, the boxer moves his head with the punch to help increase the time of impact even more. Super tankers coming into port is an extreme example. The momentum of the ship is enormous due to the large mass. So, even if the ship is moving at a slow speed, it will require a massive force to stop the ship. In order to avoid having to use the engines to slow the ship down, they turn the engines off around 15 miles from shore and use the water to slow the tanker down. The water does not seem like much, but that small force over a very large amount of time will provide the necessary impulse to reduce the momentum of the ship. Momentum Versus Energy Consider a 7.2-kg shot-put (sixteen pounds), and a .045-kg golf ball. If the shot-put was traveling at 1.0 m/s, its momentum would be 7.2 times 1.0 which is 7.2 kgm/s. The question is, how fast must the golf ball be traveling to have the same momentum? Because it has such a small mass, the speed must be much greater. Which equals an astonishing 160 m/s! That means the golf ball would travel the length of 160 meter sticks in one second!  If you were to catch both the shot-put and the golf ball in your hand with the same impact time (But not at the same time), which one would require the most force to stop? At first thought, some people might say the shot-put because it is so much more massive, and others would say the golf ball because it is traveling so fast. However, because the impact time is the same, and they both have the same momentum, the force to stop them must be the same so that the two impulses are the same. If the impact time was .10 seconds, the force to stop them would be found by using the impulse momentum equation, Which equals -72 Newtons which is approximately 16 pounds of force. (The force is negative because it slowed down an object initially traveling in the positive direction). This may seem to contradict common sense, because the next question is which one would hurt your hand the most? It seems reasonable that if they both require the same force to stop, wouldn't they also hurt the hand equally? It turns out that the answer is no! To find out why, we need to turn our attention to the energy. The shot-put's energy is found by, Which equals 3.6 Joules of energy. This means that the shot-put will do 3.6 J of damage upon impact. On the other hand, the energy of the golf ball is found by, Which equals 576 Joules! Clearly, the golf ball will do more damage to the hand, meaning it will hurt more than the shot-put. It is most likely that the golf ball would actually go right through the hand, but we will assume it does not. The interesting thing to note is that even though the stopping force is the same, the damage and pain is a lot different. To make sense of this, we again have to turn to the distance required to stop the two objects. Due to the large discrepancy in energies, it will take more work to stop the golf ball as opposed the shot-put. Again, because the force is the same, the stopping distances will be different. To find the stopping distance for the shot-put, we use the Work-Energy Theorem, The distance equals .05 meters, which is 5 cm. The stopping distance for the golf ball is found by, This equals 8 meters! That is 160 times the distance of the shot-put! This is why the golf ball would probably go through the hand, because we are not able to catch an object over a distance of 8 meters in .1 seconds. The following diagram summarizes this example.  |