Net momentum conservation for collision on frictionless horizontal surface v1i v2i Before collision m1 F on m1 from m2 During collision for t v1f m2 F on m2 from m1 v2f +x direction After collision F F From Newton s 3 rd law, on m1 from m2 on m2 from m1 Multiplying t on both sides, Ifor m1 Ifor m2 Using I-M Theorem, p p I 1 f 1i for m1 p p I 2 f 2i for m2 p p p I p I p p 1 f 2 f 1i for m1 2i for m2 1i 2i 1
System of objects We consider system of multiple objects, which we can choose any way convenient for us. For example, The system: m1, m2 and m3; however m4 and m5 are external objects! m1 m2 m3 m4 m5 2
Internal vs. External Forces Internal forces: Forces between objects within the system External forces: Forces from outside the system m1 m3 m2 m5 m4 3
Net Momentum of System of Particles (or, Objects) P p p... m v m v... net 1 2 1 1 2 2 The system: m1, m2 and m3; however m4 and m5 are external objects! v 1 m1 v 2 m2 m3 v 3 v 4 m4 v 5 m5 4
Net external force, Net external impulse Net external force : F F F... net, ext ext,1 ext,2 sum of all external forces acting on the system m1 m3 system m2 m5 m4 Net external impulse: I F t net, ext net, ext 5
Impulse-Momentum Theorem for System of Particles I P P net, ext net, f net, i Net external impulse is equal to net momentum change. Proof: use Newton s 3rd law (see textbook) Note: Internal forces do not matter for net momentum change. 6
Conservation of momentum If I net, ext 0, for example, F 0 net, ext or t is small P P net, f net, i If the net external impulse is zero, for example, if the net external force is zero, the net momentum of a system is conserved. 7
Example 1 m/s 4 m/s Before collision 3 kg particle 1 kg particle +x direction During collision After collision 1 m/s??? m/s Two objects on a frictionless horizontal surface collide as shown above. We choose two objects as a system. Find the velocity of the 1 kg object. 8
Example Before collision 1 m/s 4 m/s glue 3 kg particle 1 kg particle +x direction During collision After collision??? m/s Two objects on a frictionless horizontal surface collide and they move together after collision. Find the velocity after collision. 9
Example vi=0 Before explosion After explosion 1 m/s??? m/s 3 kg particle 1 kg particle 10 +x direction Find the velocity of 1 kg object after explosion.
Example 1: The archer An archer stands at rest on frictionless ice and fires a 0.50 kg arrow horizontally at 50.0 m/s. The combined mass of the archer and bow is 60.0 kg. With what velocity does the archer move across the ice after firing the arrow? 11
Elastic vs. Inelastic Collisions In a collision within a system of particles Before collision After collision Net kinetic energy: K K If net, i net, f If If K K K net, i net, f K net, i net, f 1 2 1 2 Knet K1 K2... mv1 mv2... 2 2 elastic collision. inelastic collision. and objects move together after collision perfectly inelastic collision. 12
Elastic Collisions in 1-Dimension Both total momentum & total kinetic energy are conserved. m v m v m v m v 1 1, i 2 2, i 1 1, f 2 2, f v1i & 1 1 1 1 m v m v m v m v 2 2 2 2 2 2 2 2 1 1i 2 2i 1 1 f 2 2 f v2i Before collision m1 v1f=? m2 v2f=? +x direction After collision m m 2m v v v 1 2 2 1, f 1, i 2, i m1 m2 m1 m2 (see text for proof) and 2m m m v v v 1 2 1 2, f 1, i 2, i m1 m2 m1 m2
iclicker Quiz In 1D elastic collision, if m1=m2, vi1=+10 m/s, v2i=0, then, after collision v1f= and v2f=. a) 5 m/s; 5 m/s b) -5 m/s; 5 m/s c) 0 m/s; 10 m/s d) 0 m/s; -10 m/s e) -10 m/s; 0 m/s Before collision After collision m1 V1i=10m/s v1f=? m m 2m v v v 1 2 2 1, f 1, i 2, i m1 m2 m1 m2 m2 and V2i=0 v2f=? +x direction 2m m m v v v 1 2 1 2, f 1, i 2, i m1 m2 m1 m2
Collisions in Two Dimensions If net external force is zero, Or P P P P net, i net, f net, i, x net, f, x P P net, i, y net, f, y, where net momentum is conserved. P m v m v net 1 1 2 2... y V1 If net kinetic energy is conserved Elastic collision not conserved Inelastic collision If moving together after collision Perfectly Inelastic collision (Kinetic energy is not conserved.) m1 v1 V2 v2 m2 x
Example 2: Collision at an intersection A 1500 kg car traveling east with a speed of 25 m/s collides at an intersection with a 2500 kg van traveling north at a speed of 20 m/s. Find the direction and magnitude of the velocity of the wreckage after the collision, assuming the vehicles stick together after the collision.
Example 1: 90 degree deflection rule in a game of pool v 2, f m m v 1,i v 1, f Assume that the collision is elastic, and the two balls have the same mass. Show that the angle between the outgoing balls is 90 degree. (No forward, back or side spin is in effect.)