p-1 Linear Momentum Definition: Linear momentum of a mass m moving with velocity v: pmv≡ Momentum is a vector. Direction of p = direction of velocity v. units [p] = kg⋅m/s (no special name) (No one seems to know why we use the symbol p for momentum, except that we couldn't use "m" because that was already used for mass.) Definition: Total momentum of several masses: m1 with velocity v1 , m2 with velocity v2, etc.. tot i 1 2 1 1 2 2ipppp mvmv≡=++= ++∑…… Momentum is an extremely useful concept because total momentum is conserved in a system isolated from outside forces. Momentum is especially useful for analyzing collisions between particles. Conservation of Momentum: You can never create or destroy momentum; all we can do is transfer momentum from one object to another. Therefore, the total momentum of a system of masses isolated from external forces (forces from outside the system) is constant in time. Similar to Conservation of Energy – always true, no exceptions. We will give a proof that momentum is conserved later. Two objects, labeled A and B, collide. v = velocity before collision, v' (v-prime) = velocity after collision. vA mA mAvA' collide mBvB mB vB' Conservation of momentum guarantees that tot A A B B A A B Bp mvmv mvmv′′=+=+ . The velocities of all the particles changes in the collision, but the total momentum does not change. Before After 3/20/2009 ©University of Colorado at Boulderp-2 Types of collisions elastic collision : total KE is conserved (KE before = KE after) superball on concrete: KE just before collision = KE just after (almost!) The Initial KE just before collision is converted to elastic PE as the ball compresses during the first half of its collision with the floor. But then the elastic PE is converted back into KE as the ball un-compresses during the second half of its collision with the floor. inelastic collision : some KE is lost to thermal energy, sound, etc perfectly inelastic collision (or totally inelastic collision) : 2 objects collide and stick together All collisions between macroscopic (large) objects are inelastic – you always dissipate some KE in a collision. However, you can have an elastic collision between atoms: air molecules are always colliding with each other, but do not lose their KE. 1D Collisions In 1D, we represent direction of vectors p and v with a sign. (+) = right (–) = left (+) vA = + 2 m/s ⇒ moving right vB = – 3 m/s ⇒ moving left Notation Danger!! Sometimes vv = speed=(always positive). But in 1D collision problems, symbol "v" represents velocity : v can (+) or (–). 1D collision example: 2 objects, A and B, collide and stick together (a perfectly inelastic collision). Object A has initial velocity v, object B is initially at rest. What is the final velocity v' of the stuck-together masses? v' vmA tot,before tot,afterAA BB A Bv0AAAABppmv mv (m m)vmv (m m)vmvvmm=B′+=+′=+⎛⎞′=⎜⎟+⎝⎠ Notice that v' < v, since mA/(mA+mB) < 1. Before After mA+mBmB (at rest) 3/20/2009 ©University of Colorado at Boulderp-3 Another 1D collision example (recoil of a gun). A gun of mass M fires a bullet of mass m with velocity vb . What is the recoil velocity vG of the gun? m tot (before) tot (after)GbGbGbp0p0MvmvMv mvmvvM===+=−⎛⎞=−⎜⎟⎝⎠ vb = 500 m/s, m = 10 gram = 0.01 kg, M = 3 kg ⇒ G0 010v 500 1 7 m/s3..=− ⋅ =− Quite a kick! This is how rockets work! Rocket fuel is thrown out the back of the rocket, causing the rocket to recoil forward. There is NO WAY to make a rocket go forward in space except by throwing mass out the back. Any other means of propulsion would violate Conservation of Momentum. (Sorry Star Trek fans, warp drive is impossible.) Incidentally, why is the barrel of a rifle so long? Answer: v = a⋅t ⇒ long barrel, more time to accelerate, bigger v Impulse To prove that momentum is conserved in collisions, we need the concept of impulse, which relates force to changes in momentum. Newton never wrote Fnet = m a. He wrote an equivalent relation using momentum: netdpFdt= Net force is the rate of change of momentum. Let's check that this is the same as Fnet = m a. pmv, p m v =∆=∆(assuming m = constant) ⇒ netpmvFmtta∆∆== =∆∆ In the special case that a constant net force is applied during a time interval ∆t , we have Before After vb > 0 MvG < 0 mMptot = 0 (both at rest) 3/20/2009 ©University of Colorado at Boulderp-4 netpFt∆=∆ or ∆=. If the force varies over time, then the correct expression is netpF t∆tfnet net,AVGipFdtF∆= = ∆∫ Definition: impulse J = net force × time netJF t≡⋅∆ ( Fnet = constant during time interval ∆t ) In general, . So we have fnet net,avgiJFdtF≡=∫ t∆ net,avgpJF t∆= = ∆ To change the momentum of an object, you must apply a net force for a time interval. The term "impulse" is usually reserved for situations in which a BIG force acts for a short time to cause a rapid change in momentum. Like a bat hitting a baseball: +x right = + directionbaseballvi fi f i fiJ p p p mv mv m v v m v=∆ = − = − = − = ∆  () Example: mbaseball = 0.30 kg , vi = – 42 m/s , vf = +80 m/s , duration of bat/ball collision = ∆t = 0.010 s What is the impulse? And what is the size of the average force exerted by the bat on the ball? Before swing:(vi < 0) mPow! It's a hit!vfm (vf > 0) After swing:vi vf pipfJ= ∆p∆v3/20/2009 ©University of Colorado at Boulderp-5 J = m(vf – vi) = (0.30 kg)(80 m/s – (– 42 m/s)) = 0.30(122) ≅ +37 kg⋅m/s (Impulse is to the right.) 37 kg m/spF 3700 N 800 lbst 0 010 s.⋅∆== = ≅∆ Bat exerts a BIG force for a short time. Proof that momentum is conserved Now finally, we are ready for the proof that momentum is conserved in collisions. We are going to show that Newton's 3rd Law guarantees that (total momentum before collision) = (total momentum after collision) We will show that when two objects (A and B) collide, the total momentum remains constant because ; that is, the change in momentum of object A is exactly the opposite the change in momentum of object B. Since the change of one is the opposite of the change of the other, the total change is zero: tot A