(t)dt I. p i. (impulse) F ext. Δ p = p f. Review: Linear Momentum and Momentum Conservation q Linear Momentum. Physics 201, Lecture 15

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1 Physics 0, Lecture 5 Today s Topics q ore on Linear omentum nd Collisions Elastic and Perfect Inelastic Collision (D) Two Dimensional Elastic Collisions Exercise: illiards oard Explosion q ulti-particle System and Center of ass Ø Hope you ve previewed Chapter 9. Review: Linear omentum and omentum Conservation q Linear omentum p p + p + p +... mv + mv + mv +... p q Impulse-omentum theorem Δ p p f p i q omentum Conservation: p f F ext (t) I (impulse) pi, if Fext 0 j Review: Collisions q Collision: n event in which two particles come close and interact with each other by force. omentum is conserved in collision: P f P i (Per Impulse approximation) Kinetic Energy of the system may or may not be conserved: Elastic: KE f KE i Inelastic: KE f KE i Review: -Dimentional Elastic Collision f f q Take v i 0: If m <<m : v f -v i, v f v i 0 (think of a tennis ball hitting ground) If m >> m : v f v i, v f v i If m m : v f 0, v f v i (demo last Thursday) v v m m m v i + v m + m m + m i m m m v i + v m + m m + m i Two extreme cases: Elastic and Perfectly Inelastic.

2 Review: -Dimentional Perfectly Inelastic Collision q Perfectly inelastic collision: fter collision, two particles have same velocity v f. q omentum in x direction: P i P f m v i +m v i m v f +m v f à v f (m v i +m v i )/ (m +m ) q Question: Is kinetic energy the same before and after? efore: KE i ½ m v i + ½ m v i fter: KE f ½ m v f + ½ m v f à KE f -KE i - ½ m m /(m +m ) (v i -v i ) <0!!! Quizzes: What is the work done in collision? Where is the lost energy? Two-Dimensional Elastic Collision q Collision can be -D, the same approach as in D works. m v i m v f m v i Ø omentum conservation P i P f (P ix P fx, P iy P fy ) m v xi +m v xi m v xf +m v xf m v yi +m v yi m v yf +m v yf Ø Elastic KE i KE f : ½ m v i + ½ m v i ½ m v f + ½ m v f Ø Three equations and four unknowns (v fx, v fy, v fx, v fy ) requires one more assumption to get full solution. Ø The rest is algebra! m v f y x Glancing Collision Target is at rest q Elastic KE i KE f : ½ m v i ½ m v f + ½ m v f Note: v v x +v y q omentum conservation P i P f (P ix P fx, P iy P fy ) m v xi m v xf +m v xf m v yi m v yf +m v yf q Three equations and four unknowns (v fx, v fy, v fx, v fy ) gain, requires one extra given condition to get full solution. y x Exercise: illiards oard q Find the angle θ of the cue ball after collision. (ssuming elastic collision, and all masses equal) Solution: Elastic: KE i KE f à ½ mv i ½ m v f + ½ m v f omentum Conservation p i p f x: mv i mv f cosθ + mv f cos5 o à y: 0 -mv f sinθ + mv f sin5 o Solve (exercise after class): cos(θ+5 o ) 0 θ+5 o 90 o θ55 o In general: The two balls always makes 90 o after collision!

3 nother Trick for illiard oard q Show that if elastic collision, and in the limit that table mass is much larger than ball mass m, θ f θ i. Keys to Solution: >>m table does not move. ll kinetic energy carried by ball. Elastic: KE f KE i à v f v i v fx + v fy v ix + v iy Normal force no force in x direction à v fx v ix è v fy v iy i.e. v fy - v iy Trigonometry: tanθ i v ix /v iy, tanθ f v fx /v fy Explosion q Explosion: single object, often at rest, breaks into multiple moving pieces within a very short period of time efore Explosion Total momentum is conserved (Impulse approximation) p f p i (note the vector form!) Kinetic Energy is not conserved! efore: v0 KE i 0 fter: KE f ½ m v + ½ m v + ½ m v +... Σ ½ m j v j >0 v m m 4 v m m v fter v 4 à θ f θ i Quick Quiz Quick Quiz before v0 before v0 after after Which of these is a possible after state? both Which of these is a possible after state? both

4 Exercise: a Simple Explosion q Find v after the string is cut. (.00Kg, ignore all frictions) q Solution: p i 0, p f v + (.00) p i 0 v m/s q Energy consideration: before: KE i 0 after: KE f ½ (6.00) + ½ (.00) 4 J Ø Quiz: Where does this 4J come from? Ø nswer: from energy initially stored in the spring For a real bomb, the energy comes from chemical energy in TNT (fter Class) Conceptual Exercise q gun of mass gun is firing a bullet of mass bullet. How does the recoil of the gun depend on the mass of the bullet? q nswer/solution: This is not an easy quiz at all! It requires a full solution. Ø Problem setting: v Gun v ullet before all at rest: p i 0, KE i 0 after p f Gun v Gun + ullet v ullet 0 KE f ½ Gun v Gun + ½ ullet v ullet E Ø Solve: v Gun ~ (E ullet ) ½ / Gun if ullet smaller, v Gun smaller Energy from gun powder if Gun >> ullet ulti-particle System and Center of ass For a multi-particle system: m, m, m,... at r, r, r,... one can define: Ø Total mass: Σ m j m + m + m +... Ø Center of ass (C) position: m r + mr + mr +... r C Ø C Velocity and cceleration v C d r C a C d v C m v + m v + m v +... m a + m a + m a +... r m r C C r m m r Exercise: Find Center of ass q Find the C for these object system. (all masses same) mx + mx + mx m 0 + m 0 + m L L x C m + m + m m q Some examples of C m y + m y + m y m L + m 0 + m 0 L y C m + m + m m Ø Now think of C as a virtual particle, it has, r, v, a 4

5 Quick Quiz: C Location q For the base ball bat below, which point is closer to the center-of-mass a. b. c. C C Quick Quiz: Dividing at C q baseball bat of uniform density is cut at the location of its center of mass as shown below. Which piece has the smaller mass? a. The left piece b. The right piece c. oth pieces have the same mass Quick Quiz: v C and omentum q It is known that at a particular moment, the total momentum of a multi-particle system is zero. Which of the following statements is true? : The system s total kinetic energy is zero : The total external force on the system is zero C: The center of mass velocity of the system is zero D: oth and C above are correct E: None of above is correct v C d r C m v + m v + m v +... p j p Dynamics of Center of ass q Impulse-omentum Theorem: F ext d p F ext d p m dv + m dv + m dv +... d v C a C The motion of C follows Newton s nd Law if only external forces are considered! 5

6 otion of C C follows projectile trajectory! (if gravitation is the only external force) 6