FZX: Pesonal Lectue Notes fom Daniel W. Koon St. Lawence Univesity Physics Depatment CHAPTER 7 Please epot any glitches, bugs o eos to the autho: dkoon at stlawu.edu. 7. Momentum and Impulse Impulse page 1
FZX, Chapte 7: MOMENTUM and IMPULSE Befoe we talk about what these wods momentum and impulse mean, let s talk about collisions. When two things collide, a numbe of things can happen. If two clouds collide, pehaps they just pass though each othe as if nothing happened. If two billiad balls collide, they ecoil fom each othe with nealy the same elative speeds as they appoached each othe with. If two magnets collide, they may stick to each othe afte the collision. It is impossible to know befoehand what the final velocities will be without having an intimate knowledge of how the two objects eact to each othe while they ae in contact. In the wods of physics, we need to know what foces they will exet on each othe duing the collision. Well, usually we don t know these foces. We have not intoduced any quantities that allow us to distinguish between a ubbe ball and a ball made of modeling clay. The inteaction between two objects is sometimes so shot that these foces ae vey lage and vay quickly with time. This just about ules out using Newton s Second Law to solve fo the [nonunifom] acceleation duing the collision. So how do we analyze such a poblem? Newton s Thid Law helps us out hee. While the two objects ae in contact, we don t know what foces they exet on each othe, but we do know that they ae equal and opposite. If we conside the two objects sepaately, of couse, we cannot cancel out these foces (since they ae on diffeent objects!), but if we conside the two objects togethe, and teat them like a single object, then we can. If no net outside foce is acting on the system of two (o moe) objects, then the system will not acceleate. That s small consolation fo us, because when we go back to the two sepaate objects, both acceleate. If we ewite Newton s Second Law fo a single paticle as Δv F = ma = m, Δt then we can see that if the net foce on a paticle is zeo, then m v = 0 o m v = constant. We will call this new quantity the momentum, p, and efe to this pinciple as the Pinciple of Consevation of Momentum: p = mv [Momentum] in units kg. m/s [Units of momentum] If F = 0, then p = mv is conseved. [Consevation of momentum] Now, this is not only tue fo a single paticle -- that s just Newton s Fist Law -- but also fo a system of paticles. The total momentum of the system is unchanged if no net foce acts on it, although momentum may be tansfeed fom one object to anothe. Although momentum was egaded by both Newton and Descates as the pincipal quantity of motion, the physics community has not seen fit to come up with a sepaate name fo the units of momentum (like newtons, joules, etc.), as they have done fo the units of foce. That s one less new name to memoize, but it does make it hade to ecognize if you ve got the ight units. Momentum is a vecto quantity. That means that if we have a collision in two dimensions, the momentum components in the x-diection and in the y-diection must both be conseved. Fo a system of two o moe paticles, If AND F = 0, then p... 1 x + p2x + = p1x + p2x +... p... 1 y + p2 y + = p1 y + p2 y +... [Consevation of momentum in component fom] whee the quantities on the left-hand side ae the initial values, and the pimed quantities on the ight-hand side ae the final values. page 2
Okay, but what about the specifics of the collision? This equation, you will find, does not give you enough infomation to detemine the final velocities of all the colliding objects. These depend, again, on the specifics of the foces acting between the objects duing the collision. In ode to figue out the final speeds we need to know the paticulas. Thee ae two specific kinds of collisions that set the limits fo what the final velocities ae in MOST collisions. If the two objects ecoil at the same elative speeds that they came in with, one can show that they have conseved kinetic enegy. This is called a totally elastic collision. KE 1 + KE2 +... = KE 1 + KE2 +... [Totally elastic collision] If the objects stick togethe afte the collision, then we have a totally inelastic collision. v = 1 v 2 [Totally inelastic collision] Most actual collisions ae somewhee inbetween these two extemes. In most collision poblems you will encounte in physics, you will be told (diectly o indiectly) that you have eithe a totally elastic o totally inelastic collision, o else you will be given one of the final velocities. Use this infomation, but most impotantly, since it is a collision, don t foget to apply Consevation of Momentum! page 3
page 4
IMPULSE: Okay, it s nice to know that you momentum plus the momentum of the guy in the tuck who just ea-ended you is conseved, but moe to the point, you want to know by how much did that othe dive change you momentum. Afte all, this is what distinguishes between you having a mino scape on you fende o getting whiplash o wose. If you know the initial and final velocities of eithe of the objects colliding, then you can figue out how much momentum it picked up o lost, and the diection of the change in momentum. This change in momentum of an individual object is called the impulse, and the impulse on that object will exactly equal the negative of the impulse on the othe object, since momentum is conseved. Impulse must have the same units as momentum. It is also a vecto, and we can wite it as I Δp = [ Impulse ] Reveting to Newton s Second, we can wite this as I = Δp = mδv = ma avg Δt = F avg Δt. This allows us to estimate the foce if we know how long the two objects ae in contact. Again, let me say that the foce is non-unifom, so this is at best a cude estimate of the foce, teating it as if it wee unifom duing the time peiod t. mδv = F avgδ t [ Impulse Appoximation ] Okay, you e pobably asking, how do I tell how long two objects ae in contact, if they act ove a vey shot time, as is tue fo a bat hitting a baseball, a kicke kicking a football, a golfclub hitting a golfball, an automobile hitting a bick wall? We can tell if we know how much one of the objects is squashed duing the collision, which we can do by high-speed photogaphy, o by measuing how much we can compess a baseball with a given foce. If we conside a ca hitting a bick wall, a collision which diffes fom ou spots examples in that the ca does not sping back to its oiginal shape, we can measue how quickly the collision took place by measuing how much the ca s font compatment collapsed. We can then teat the ca s motion unde the influence of the wall s contact foce as a kinematics poblem in unifomly acceleated motion, whee the distance the ca tavels (duing the collision) equals the length by which the ca is scunched. page 5
page 6
page 7