Chapter 9 Linear Momentum and Collisions Momentum 2 Linear momentum of object: mass velocity Magnitude: p=mv, Direction: same as velocity Quantity of motion, ability to acting force Newtons second law of motion (original edition) The changing rate of momentum of an object is equal to the net force applied on it. Impulse 3 Change in momentum is the integral of the net force over the time, the integral is called impulse Integral form of Newtons second law momentum ~ impulse energy ~ work Eq. is called impulse-momentum theorem 1) any force3)2) vector Projectile problem 4 Example1: Object m is fired with v at angle 45to the horizontal, no air friction. Determine: a) Initial momentum b) Final momentum c) Impulse during the motion Solution: a) x y v 45 b) same magnitude different direction c) Conical pendulum 5 Example2: Known: m, L, , v, T. Determine the impulse of tension and gravity respectively during one-half period. O m L Solution: FTFT G So: F T is not a constant force! Direction of these vectors Collisions and impulsive force 6 Concept of momentum and impulse is useful when dealing with collision problems Key points: 1) Time interval is usually very short 2) Interaction of collision, or impulsive force is changing fast and may become very large The effect is shown by its impulse 3) Impulses of other forces can be ignored Bend your knees 7 Example3: A 70-kg person jumps from h=3.0m and lands on firm ground. a) Calculate the impulse experienced; b) Estimate the average force by the ground if he is stiff- legged (s=1.0cm); c) average force with bent legs (s=50cm). Solution: a) Landing speed Impulse b) c) impulsive force x Washing a car 8 Example4: Water is jetting at rate of R=1.5kg/s with v=20m/s, and is stopped by a car (no splashing back). What is the force exerted on the car? Solution: In a short time interval dt, How much water hits on the car: dm=Rdt Change in momentum: dp=vdm=Rvdt Collision: ignore gravity, only force by car What if the water splashes back? Falling rope 9 Example5: One end of a hanged rope just touch the table, then it is released. Prove: Normal force acting on table is always 3 times of weight of that part already on the table. Solution: Linear density , falling height h Falling speed m Mass of part already on table m = h In time interval dt,dm = vdt Impulse Fdt = dmv = v 2 dt F = v 2 = 2gh =2mg F mg dm v Momentum of a system 10 Consider a system of n interacting particles f 1n f n1 f i1 f 1i FiFi F1F1 mimi m1m1 FnFn f ni f in mnmn Total momentum of the system Change in total momentum Internal forces are always in pairs Only external forces can change the total momentum! Conservation of momentum 11 If the net external force is 0, then This is the law of conservation of momentum: When the net external forces on a system is zero, the total momentum remains constant. Or: The total momentum of an isolated system of bodies remains constant. 1) isolated2) system3) ignoring forces 12 Rifle recoiling Example6: Calculate the recoil speed of a 5kg rifle that shoots a 10g bullet at v=700m/s. Solution: Total momentum is conserved The rifle moves back and applies a recoil force 13 Billiard ball collision Example7: A billiard ball with v in the +x direction strikes an identical resting ball. Directions after collision are shown in figure, what are the speeds? Solution: conservation of momentum 45 x o y component form: kinetic energy is also conserved 14 30 m v Conservation in component form Example8: Bullet (20g) hits into hanging ball (980g) with v=400m/s. Determine the speed after collision. M Solution: No net force in horizontal Horizontal component P x is conserved T (M+m)g If, but x component of momentum is conserved 15 Homework A gun carrier M moves on a frictionless incline, its speed reduces from v to 0 after shooting a canon-ball m in the horizontal direction. Is the total momentum of system (M and m) conserved in this process, and why? Find out the speed of canon-ball. M v mv 16 Distance traveled Example9: Two objects start from rest, what is the distance traveled by m 2 when m 1 reaches the ground? No friction. a b m1m1 m2m2 x o Solution: conservation of momentum in horizontal where So distance traveled by m 2 17 Challenging question Question: Someone sits on the top of an ice half sphere on the ground, and then slips down. At what angle will he leave the sphere? No friction. top Leave the half sphere: N = 0 Thinking: ? Relative speed & inertial frame Conservation of momentum and mechanical energy Velocity transformation m M Elastic collisions 18 Conservation of momentum (1-dimension) It is called an elastic collision If total kinetic energy is conserved in a collision Ideal model in macro world Relative speed: equal but opposite head-on collision 19 Final speeds Special cases: 1) Equal masses 2) Target m 2 at rest 20 Baseball batting Example10: A baseball with speed 40m/s is hit by a bat with speed 30m/s in the opposite direction. Determine the speed of baseball if the collision is elastic and m bat >> m ball. Solution: final speed of ball Or in the frame of bat: The ball moves in 70m/s, and rebounds in the same speed Transform to the frame of ground, its 100m/s. 21 Slingshot effect Example11: Spacecraft Voyager II approaches the Jupiter, it rounds the planet and departs in the opposite direction. What is the speed after this slingshot encounter? ( v S =10.4km/s, v J = - 9.6km/s ) Solution: Like an elastic collision speed after slingshot: Inelastic collisions 22 It is called an inelastic collision If kinetic energy is not conserved in a collision Restitution coefficient e=1: elastic collision e