One Diensional Collisions These notes will discuss a few different cases of collisions in one diension, arying the relatie ass of the objects and considering particular cases of who s oing. Along the way, I ll try to gie exaples appropriate to each type of collision. Ultiately, a force is a force is a force, and there s no FUNDAENTAL difference between the forces inoled when a ball and the Earth interact to ake the ball fall to the ground and the forces inoled when the Earth and the ball interact to ake the ball bounce. But in the latter case, the forces are BIG and interact oer a SHORT tie (we hae called these ipulsie forces. This is typically what is eant by the word collision and it does ake sense to kind of group the off by theseles. So we will. What is nice about the fact that the forces are BIG and act oer a SHORT tie is that they are often uch ore influential than all other forces for exaple, the forces inoled in the collision of a ball with a bat or when a firear is fired or when a neutron collides with a Uraniu nucleus (in a reactor or a bob are typically UCH greater than the weights of the objects. So it is usually the case that the effect of the other forces is inial during the collision. So, in collisions we will consider that there are no net, external forces. Then fro the Equation: r r PTOT FNET, EXT 0 ( t we can conclude that the total oentu does NOT change that is, the total oentu before the collision will be the sae as the total oentu after the collision. So consider the siplest collision iaginable that is not triial two objects oing along a line. (In general, ore objects in ore than one diension ake the proble ore coplicated, but do not change the physics inoled the sorts of conclusions we will draw do not depend on the collision being one-diensional. Consider the drawing: Before V V After It shows an object of ass oing at speed colliding with an object of ass oing at speed V. After the collision, the first object of ass is oing at speed and the second object of ass is oing at speed V. Despite the drawing, we ake no assuptions about which object is ore assie (in fact, in the exaples we ll consider all different possibilities. Knowing that the total oentu is consered, we can write that the oentu before is equal to the oentu after:
PTOT, Before PTOT, After ( + V + V ' In these sorts of probles, you re interested in what happened after the collisions that is, you want and V. As things stand, we hae only one equation and two unknowns. To sole for AND V, we need another equation (or so say the atheaticians. So we further classify interactions in ters of what happens to the kinetic energy of the two objects. In general, soe fraction of the kinetic energy will be lost, anywhere fro none (a perfectly elastic collision to the ost possible, gien that oentu ust be consered (a perfectly INelastic collision. We will consider the two extrees and along the way I ll say a few words about the general case. Perfectly Inelastic collision In this case, the two objects stick and oe at the sae elocity after the collision: V. It is straightforward to find the final elocity using Eq. : + V V ' (3 + An exaple of this kind of collision would be ud splattering on a car window or a theral neutron colliding with a uraniu nucleus or a eteor colliding with the Earth. As we will see, there is a loss of kinetic energy in this type of collision for the ud on the window, the splattering takes energy and there is soe heat generated. In the uraniu nucleus, the lost kinetic energy causes the nucleus to start sloshing around until it breaks in half (this is called fission. With a eteor colliding with the Earth, the lost kinetic energy showed up as a assie explosion that caused the extinction of the dinosaurs. As an exaple, suppose we ARE talking about the collision of a assie, house-sized eteor with the Earth. Suppose that the Earth and the eteor collide head on, going about the sae speed. (In fact, the eteor will be going about 40% faster, as you can calculate yourself see p. 60 and 93 of your text. Also, there s no way to tell yet whether the collision was head-on, fro the side or what. But we can proceed nonetheless. That is, we assue V - (sae speed, but in opposite directions. Then Eq. 3 tells us that the final speed is: + V V ' V (4 + + Considering that the eteor has a ass of aybe 000 tons ( 0 6 kg and the Earth a ass of about 0 5 kg, it s pretty clear that the final speed will be about equal to the speed the Earth was going before the collision (uch like a fly hitting a windshield, only ore so. We won t do the calculation here in the interest of tie, but a firear can be considered an inelastic collision in reerse the kinetic energy is created by the exploding gasses rather than lost and the oentu is consered when the recoil of the firear balances the oentu of the bullet. A couple of other exaples that you should do yourself are: Suppose one object (say is initially not oing (V0. Then the final elocity is: V ' (5 + +
And suppose further that the stationary object is uch ore assie than the oing object ( << this is a tiny piece of clay colliding with the wall of a building. Then the final elocity is: ' V ' << (6 That is, as we know, the cobined object will oe away at a elocity UCH less than the initial elocity of the saller object. On the other hand, suppose that the stationary object is uch LESS assie ( >> this is a fly being hit by a car. Then Eq. 5 becoes: ' V ' (7 + That is, the cobined object goes about the speed of the initially oing object (as would be expected. We re really done here (since we know how to calculate the final elocities of the colliding objects but we can consider a couple of other ites while we re here. One is the idea of the center of ass of the syste. That s a little easier to deal with since we only hae two objects. We e concluded in class that P TOT TOT CO, so it s pretty easy to sole that for the center of ass of the syste: PTOT + V CO (8 TOT + where I e used the oentu before the collision. If one of the objects has a UCH larger oentu than the other, we can see that the elocity of the center of ass will be about equal to the elocity of that object (the Earth and the eteor, for exaple. But note that if the total oentu is consered, then the elocity of the center of ass ust be the sae (if P TOT TOT CO, and P TOT is constant, then CO ust be constant and so we should get the sae answer if we calculate CO after the collision. So let s do it: PTOT + V ' + + V CO (9 TOT + + + So, as adertised, we see that the elocity of the center of ass is the sae before and after the collision. But een ore, we see that the elocity of the center of ass is the sae as the speed of the cobined particle after the collision (which akes sense if there s only one thing, the elocity of the center of ass ust be the sae as the elocity of that thing. The second thing to consider is the kinetic energy lost in an elastic collision (appropriate since we will next consider elastic collisions in which the kinetic energy IS consered. This is typically done by considering the fraction of kinetic energy lost and this is historically labeled Q : KEi KE Q KE i f + V V ' + V ' + V + V We can factor the ters in the nuerator in the following way: (0
( ( + + ( V V '( V V ' + V V ' + ( Fro conseration of oentu, we know that: ( ( V V + V + V ' so that ' ( And so the nuerator can be written: ( ( + + ( V V '( V + V ' ( ( + V V ' (3 And so the alue of Q can be written: ( ( [ V + ( V '] Q (4 + V This is a ery general expression that is, we haen t said anything about what kind of collision is inoled. We ll use it later to consider elastic collisions, but for now we ll consider what happens in an inelastic collision. In that case, we hae that V and is gien by equation 7. So, the expression for Q becoes: + V ( V ( V Q + + (5 + V + V For the exaples discussed earlier, we can calculate Q. For the eteor striking Earth, we hae -V: Q + + (6 ( ( 4 ( Since <<, we can see that the fraction of kinetic energy lost is tiny. This akes sense the VAST ajority of the kinetic energy is the kinetic energy of the Earth, which really isn t affected. But it turns out that four ties the kinetic energy of the eteor is dissipated in the collision: K lost Q K 4 i 4 + V ( ( + 4 ( 4 Finally, fro Equation 6 we see that the axiu loss of kinetic energy occurs when the asses are equal and the objects are oing in opposite directions. This is the reason that particle physicists sash atos by running the into one another going in opposite directions this gies the axiu aount of kinetic energy to ake new particles. Perfectly Elastic Collision In this type of collision, the oentu is still consered, so we hae Equation ( as one equation to sole for and V. And we know that kinetic energy is consered, so this is another (7
equation. But we can short cut a LOT of algebra because we already did it. Consider Equation (4. This is the general expression for Q. But for a perfectly elastic collision, we know that Q 0. So we can get the second equation for and V fro there. If Q 0, then it ust be that either (both objects oe at the sae speed in the sae direction and therefore NEVER collide OR: ( V + ( V ' 0 (8 which gies us two equations to sole for and V. So let s. I can sole Equation (8 for V : V ' V + (9 Substitute this into Equation ( to get: ( + V + V + ' (0 Soling this for, we get: ( + V ( And plugging this into Equation (9, we get: ( V + V ' ( Again, we re basically done, as we hae the elocity for both particles after the collision in ters of the initial elocities and the asses. But we can consider a few exaples: Elastic collision one object initially still: Suppose one object is standing still (V0. Then the elocities reduce to: V ' ( If the asses are equal (, then these elocities becoe: V ' ( 0 In this case, the speeds are switched (the initially otionless object oes at the speed of the oing object and the initially oing object stops. This describes a gae of pool the cue ball oes toward the object ball, strikes it and stops as the object ball goes into the hole. (It also shows the answer to one of your HW probles. If the otionless object is ery assie ( >>, then: (3 (4
V ' ( ( ( + << In this case, the sall object reerses its elocity (the sae speed in the opposite direction and the large object oes a little (just enough to consere oentu. An exaple would be a superball hitting the Earth. Finally, suppose the initially otionless object is ery sall ( << : V ' ( ( ( The large oing object continues pretty uch unaffected and the sall otionless object bounces at twice the elocity. If a train with a cowcatcher hits a car, the car will bounce away at a speed faster than that of the train (although this is NOT likely to be perfectly elastic. Now suppose that the objects trael at the sae speed, but in opposite directions: Elastic collision objects trael in opposite directions: Traelling at the sae speed but in opposite directions iplies -V: V ' ( + ( V + V Suppose the asses are equal (: 3 ( + ( 3 3 V ' V ( + ( 3 ( + + 3 3 In this case, both objects bounce back in the opposite direction with the sae speed. Suppose one object is uch ore assie than the other (it doesn t atter which, as you should proe suppose << : 3 3 3 3 V ' V Here (again the assie object is largely unaffected by the collision, but the light object bounces back at 3 ties the speed. This describes an experient you can perfor balance a tennis ball or a superball on a basketball and then drop the together. The effect is draatic. (5 (6 (7 (8 (9