Physics I 95.141 LECTURE 16 11/03/10

1. Physics I95.141LECTURE 1611/03/10

2. Outline Impulse Conservation of Momentum and Energy Elastic and Inelastic Collisions What do we know? Units Kinematic equations Freely falling objects Vectors Kinematics + Vectors = Vector Kinematics Relative motion Projectile motion Uniform circular motion Newton’s Laws Force of Gravity/Normal Force Free Body Diagrams Problem solving Uniform Circular Motion Newton’s Law of Universal Gravitation Weightlessness Laws Work by Constant Force Scalar Product of Vectors Work done by varying Force Work-Energy Theorem Conservative, non-conservative Forces Potential Energy Mechanical Energy Conservation of Energy Dissipative Forces Gravitational Potential Revisited Power Momentum and Force Conservation of Momentum Collisions Impulse

3. Collisions and Impulse • Over the course of a collision, the inter-object Forces change very quickly. • Example: a serve in tennis….. • We can think of this as a spring system, with the compression/extension of the ball/strings occurring over a small fraction of a second (~5ms)

4. Collision and Impulse • From Newton’s second law, we can write: • This integral is known as the Impulse

5. Collisions and Impulse • The impulse of a Force is simply the integral of that Force over the time the Force acts.

6. Example • Imagine the force exerted by a tennis racket on the ball during a serve can be approximated by the F vs time plot below. What is the impulse acting on the .056 kg ball? What is the speed of the serve? Force (kN)

7. Shaken, not stirred……

8. Bond, James Bond • Using Energy, can we explain why the window doesn’t break when they push off, but does when they come back to it?

9. Bond, James Bond • Using Impulse, can we why the window breaks? • Assume the push-off takes 0.5 seconds, so that Bond goes from 0m/s to vo in 0.5s. • On the return, however, Bond and Wai Lin brace their legs, and they are slowed to a stop in 0.05 seconds.

10. Bond, James Bond • Calculate average Force for each case (push-off and impact)

11. Conservation of Momentum • In the previous lecture we discussed the quantity of momentum • The change of momentum of an object can be related to the net Force on the object • In a collision, momentum is conserved, as long as no external forces act on the system • Impulse

12. What about Energy? • In the case of the fender bender where the cars lock bumpers and travel on together, is kinetic energy conserved? • Remember, Car1 (1000kg) traveling at 10m/s hits Car2 (1000kg), which is at rest, and their bumpers lock together.

13. Conservation of Energy • Total energy is always conserved. • In collisions, sometimes we can say that not only is total energy conserved, but kinetic energy is conserved. • During a collision, for a split second, some or all energy is stored in elastic potential energy. • But this energy is quickly returned to either thermal or kinetic energy (or both) • Two hard elastic objects (billiard balls) usually end up with the same total kinetic energy • When this happens, the collision is referred to as an elastic collision.

14. 1D Elastic Collisions • We must now consider both conservation of energy and momentum. • In the last section, we had to give the masses and 3 out of 4 velocities. • If we know the collision is elastic, then we only need to know 2 out 4 of the velocities.

15. 1D Elastic Collisions Example • Ball 1 is traveling with a velocity of 10 m/s and Ball 2 with a velocity of 3 m/s. What are the final velocities of the balls, if they have the same mass? 3m/s 10m/s

16. In General • Say you have two masses (mA and mB), each traveling with initial speeds (vA and vB).

17. Interesting…. • For any elastic collision in 1D, the relative speed of the two objects after the collision is the same as it was before the collision, but the direction opposite! • This is a simpler way of writing conservation of Energy for 1D!

18. Elastic Collision, Equal Masses • If you start with two mass (mA = mB).

19. Unequal Masses, Target at Rest • If you start with two mass (mA and mB) with mB at rest. • What happens if mA is much more massive than the target? • What if target is much more massive (mA << mB)

20. Newton’s Cradle • Given what you now know about elastic collisions, you should be able to explain this:

21. Inelastic Collisions • A collision where Kinetic Energy is not conserved is known as an inelastic collision. • Technically, all collisions are inelastic, since there is always some energy that is converted to heat, even when we model these collisions as elastic. • A collision/process is inelastic when… • Some of the mechanical energy is converted into thermal or potential energy or… • In the case of explosions, when potential energy (chemical or nuclear, for example) is converted into kinetic energy

22. Newton’s Cradle, revisited • Suppose the period of motion for this Newton’s cradle is 0.5 s, and the height the 100g ball swings to decreases by 1mm each swing. How much thermal energy is generated each period? What is the power is dissipated?

23. Perfectly Inelastic Collisions • If two object stick together after a collision, this is known as a perfectly inelastic collision. • Let’s revisit our car crash again. • 1000kg car, travelling 10m/s hits a car at rest and their bumpers lock (perfectly inelastic collision)

24. Dirty Harry

25. What is the final speed of the dead guy? • mbullet=20g • vbullet=405m/s • mbadguy=70kg

26. Ballistic Pendulum • A device used to measure the speed of a projectile. m h M M+m vo v1

27. Ballistic Pendulum m M+m M vo v1

28. Ballistic Pendulum M+m h M+m v1

29. Ballistic Pendulum • If the projectile mass is 10g and the pendulum mass is 3kg, and the pendulum swings to a height of 5cm, what is the velocity of the projectile before the collision?

30. Summary • Previous Lecture: Conservation of Momentum • Today’s Lecture: Cons. Of Momentum plus Energy • Elastic Collisions (1D) • Inelastic Collisions (1D) • Perfectly Inelastic Collisions