Lecture 12: Momentum

Transcription

1 Lecture 2: Momentum So far, we ve used ewton s Laws to predict the motion of a wide variety of systems However, in many cases we don t know exactly what forces are acting, or exactly for how long Is there any hope of learning about how objects move under these circumstances? To see, let s rewrite ewton s Second Law: dv = ma = m d t d ( mv) = = (actually, ewton wrote it in this form!)

2 The quantity p is called the linear momentum of a particle ow consider a system of particles interacting with each other, but with no external force: 2 There are similar force pairs between any two particles

3 The net force acting on the ith particle is: i = ij = j= i With this, we can determine the rate of change of the momentum for the entire system: = i= i = = ij i= j= j= i= ij

4 But ewton s Third Law tells us that the interaction between two objects results in forces that are equal in magnitude but opposite in direction. In other words: ij = ji Thus we find: = = = ij ij ji i= j= j= i= j= i= = ij = i= j= This means that -- the total momentum of the = 0 system never changes

5 Since the total momentum of the system never changes, we say that momentum is a conserved quantity ote that we made no assumption about the nature of the interactions between the particles Also, since momentum is a vector, we really have three conserved quantities: p x, p y, and p z

6 Conservation Laws in Physics Knowing that momentum is conserved allows us to analyze motion in cases where we don t know the exact nature of the forces acting between particles Physicists believe that this conservation law is always true even in the situations where the approximate nature of ewton s Laws is revealed Why are we so sure?

7 The reason is that, around the turn of the century, Emmy oether proved a theorem that is part of the core of modern physics: or every symmetry in the laws of physics, there will be a conserved quantity As long as physics is symmetric with regard to translation (i.e., the laws are the same in Tucson, ew York, Mars, the Andromeda Galaxy, and anywhere else), momentum must be conserved You might be able to think of a few other symmetries which should have associated conservation laws we ll encounter a couple more this semester

8 Impulse Looking again at ewton s Second Law in the form that ewton wrote it: = = d ( mv) We can rearrange this to find: ( ) = d mv = mdv Only if mass is constant! By integrating, we find: ( ) = mdv = m v v = m v f i

9 The quantity is called the impulse, and written as ote that J is a vector quantity What can the concept of impulse tell us about the interaction between two objects? Consider an arbitrary, time-dependent force between two objects: 00 J orce() Time(s)

10 What if we wanted to find the average force over the time interval? We could take a set of measurements at different times, and take the numerical average: avg = i= ( t ) i But that s not exactly right, since we don t know what the force is in between the times we measure it. If we assume that the force remains constant between measurements, this becomes: ( ti )( ti+ ti ) ( ti ) ti i= i= avg = = t t t i= ( ) i+ i

11 To get the exact answer, we need to reduce the time interval between measurements to an infinitely small value. Then we find: avg t t f ( t) i = = t J t So we see that the impulse is related to the average force between the two objects and the total time over which the force was acting The word impulse implies a short time, and that will be true for most (but not all!) of the situations in which we use it

12 Example : Air Bag Cars now come equipped with air bags that inflate in front of the driver when a front-end collision is detected. Given that the driver experiences the same change in momentum regardless, why are these useful? Even though the impulse isn t changed by the presence of the airbag, the time of the collision is greatly increased The driver s head is in contact with the airbag for about 00x longer than it would be with a hard object like the steering wheel or windshield The force on the driver is reduced accordingly, and serious injury is more likely to be averted