Impulse and Momentum

1. Impulse and Momentum Chapter 7

2. During collisions 1) force magnitudes change rapidly 2) forces act only briefly. maximum force average force Collisions are analyzed using the concepts of impulse and linear momentum.

3. Impulse vector is equal to the product of the average force acting on an object and the time interval during which the force acts. impulse magnitude is also equal to the area under the F vs. t curve. Impulse = (average force)(time interval) Units Impulse vector The impulse vector direction is the same direction as the force vector direction.

4. Linear momentum vector is equal to the product of the object’s mass times its velocity vector. Linear momentum vector Units The linear momentum vector and the velocity vector are in the same direction.

5. final momentum initial momentum net impulse Impulse - momentum theorem An object's momentum change is equal to the net impulse acting on the object. Momentum units are equivalent to impulse units.

6. Impulse - momentum theorem: brief derivation Newton’s 2nd law

7. Example 2 Rain drops Rain drops moving downward at 15 m/s stop when they hit the roof of a car. What is the average force needed to stop 0.6 kg of rain in one second? average force of 9 N on 0.6 kg of rain impulse on rain is +9 Ns average force +9 N on the rain -9 N on the roof

8. Example 3 Hailstones Versus Raindrops Unlike rain, hail usually bounces off the roof of the car. Would the force on the hail be smaller than, equal to, or greater than the force on the rain for the same mass? assume same v magnitudes Rebounding required twice as much momentum change and that required twice as much force.

9. System "The collection of objects being studied is referred to as the system". The system concept divides the universe into two parts. 1) things that are inside the system 2) things that are outside the system.

10. Internal and external forces • Internal forces: • forces that objects inside the system exert on each other. • External forces • forces that objects outside the system exert on objects inside the system.

11. The “system” defines what is inside and what is outside. External force The system Internal forces External force

12. The “system” defines what is inside and what is outside. External force The system External force

13. Total linear momentum of a system To calculate the total momentum vector, add the momentum vectors for everything in the system. To calculate the change in the total momentum vector, add the change in the momentum for everything in the system.

14. only external forces can change the total linear momentum Impulse - momentum theorem for a system All the internal forces are mutual forces between pairs of objects inside the system. The sum of the equal and opposite internal forces inside the system is zero.

15. Principle of conservation of total linear momentum If the net external force is equal to zero, then the total linear momentum vector for the system remains unchanged. Internal forces can transfer momentum from one internal object to another, but internal momentum transfers can't change the total momentum.

16. Strategy: Principle of Conservation of Linear Momentum • Decide which objects are included inside the system. • Identify the internal forces and the external forces. • If the net external force is equal to zero... • Total linear momentum will remain unchanged. • Final total linear momentum of the system is equal to the initial total linear momentum. • Remember that momentum is a vector and that vectors have directions. Since the net external force is either negligible or zero during collisions, the total linear momentum is always conserved during collisions.

17. Balanced external forces: gravity and supportTherefore, total linear momentum is conserved. Collision example with a 2-car system m2=92000 kgv2=1.3 m/s m1=65000 kgv1=0.8 m/s vf = ?

18. Collision example with a 2-car system m2=92000 kgv2=1.3 m/s m1=65000 kgv1=0.8 m/s vf = 1.093 m/s Internal forces transferred momentum from car 2 to car 1.

19. Example 6 Ice Skaters Starting from rest, two ice skaters push against each other. External friction force is negligible. External gravity forces are balanced by external support forces. Therefore the net external force is zero and the total linear momentum is conserved. The 54 kg woman moves away with a speed of +2.5 m/s. The man has a mass of 88 kg. What is the speed of the man?

20. Total linear momentum is conserved Initial velocities are both zero. 54 kg 88 kg The man moves in the negative direction.

21. Example 8 A Ballistic Pendulum A 0.01 kg bullet hits and sticks in a 2.5 kg block of wood. The block swings to a maximum height of 0.65 m above its initial position. Find the initial speed of the bullet. Two events: collision followed by an upward swing System for analyzing the collision: bullet and block

22. Analyzing the collision The collision forces are internal forces. net external force = 0 Total linear momentum of the bullet and block system is conserved during the collision. Two velocity values are unknown. We need a second equation so that we can solve for one of these unknown velocities before can we put numbers in this equation.

23. vtop=0 h=0 Analyzing the swing Gravity and string forces act on the bullet and block during the swing, but only the conservative gravity force does any work. Total mechanical energy is conserved during the swing. at bottom of the swing

24. Total linear momentum is conserved during the collision because the net external force acting on the bullet and block was equal to zero.

25. Summary Collision friction force friction is only an internal force so linear momentum is conserved non-conservative friction force does work so mechanical energy is not conserved Swing gravity and support forces unbalanced gravity force is external so linear momentum is not conserved only conservative gravity force does work so mechanical energy is conserved

26. Collisions in two dimensions If the net external force is zero, then

27. (use vector addition using components)

28. Center of mass Weighted average location for the total mass of a system of objects. Velocity of the center of mass can only be changed by external forces from outside the system. For a two-object system

29. Elastic and inelastic collisions Elastic collision -- The total kinetic energy of the system remains the same. Inelastic collision -- The total kinetic energy of the system changes. Completely inelastic collision -- The objects stick together after colliding.

30. The End