1. Momentum and Its Conservation Chapter 9

2. Properties of a System Up until now, we have looked at the forces, motion and properties of single isolated objects. In the following chapters (9,10,11), we will look at the properties of a system of objects and how they interact. Most of the time we will be looking at before and after scenarios

3. Momentum Momentum measures the strength of an object’s motion. Momentum (p) depends on an object’s mass and its velocity. More mass = more momentum More velocity = more momentum

4. Momentum Momentum (p) = mass x velocity p = mv Momentum is a vector quantity! Momentum acts in the same direction as the velocity. Unit = kg m/s –Do not confuse with a Newton

5. Impulse and Changing Momentum To change the momentum a force is required but is not the only factor in changing momentum To change the momentum an Impulse is required. Impulse is the product of the Applied Force and the Time Interval in which the force acts.

6. Impulse and Changing Momentum Impulse = F  t Impulse is a vector quantity (it is in the same direction as the force). Unit = Ns or kg m/s (same as momentum) How can we have a Large Impulse?

7. Two Ways to Have a Large Impulse 1. LARGE FORCE over a small time interval 2. Small Force over a LARGE TIME INTERVAL

8. Impulse From a F vs. t Graph We can find the Impulse from a Force vs. Time Graph. The Impulse will equal the Area under the curve of a F vs. t graph. If the Area is a rectangle use A = lw If the Area is a triangle use A = ½ bh

9. Impulse-Momentum Theorem An Impulse causes a change in Momentum. (If the mass remains constant then the velocity of the object must change.) F  t =  p = m  v Note: Another way of expressing Newton’s 2 nd Law

10. Angular Momentum Objects traveling with a linear momentum (p) are affected by the mass of the object and the velocity Rotating objects have an angular momentum (L). An object’s angular Momentum depends on how the mass is distributed around the axis and its angular speed!

11. Angular Momentum L = I  Similar to p = mv The Impulse Momentum Theorem also applies to Angular Momentum (L).   t =  L = L f - L i

12. Conservation of Momentum Newton’s 3 rd Law applies to momentum as well as forces. Momentum will be conserved in collisions. –Conservation means the total amount remains constant!

13. External Forces and Internal Forces An External Force is a force that is acting outside your system. It may be acting on your system or have no affect on your system An Internal Force is a force that is acting within your system between two objects inside your system How you define your system determines if the forces are internal or external

14. Closed Isolated System In order to observe the conservation of momentum we need to have a Closed Isolated System (CIS). In a Closed Isolated System: 1.You must have a collection of objects 2.Nothing enters or leaves the system 3.There are no NET EXTERNAL FORCES acting on your system

15. Conservation of Momentum If no outside forces are acting on a closed isolated system of objects, then the total amount of momentum of the system remains constant. p i = p f We observe the conservation of momentum in Collisions!

16. Conservation of Momentum in Collisions Momentum can be conserved in: One Dimensional Collisions Two or Three Dimensional Collisions Elastic Collisions (1 or 2 dimensional) Inelastic Collisions (1 or 2 dimensional)

17. Elastic and Inelastic Collisions To completely define elastic and inelastic collisions requires energy considerations (Chapter 11). Since we have not gotten that far yet, we will say for now; In an elastic collision the objects separate after the collision In an inelastic collision the objects stick together after the collision

18. Conservation of Momentum Problems (1-D Collisions) 1. Count the number of objects in your system before and after the collision. 2. Substitute “mv” for every “p” in the equation. 3. Put the numbers in and solve for your unknown.

19. Conservation of Momentum in a 1-Dimensional Elastic Collision p i = p f 1. p Ai + p Bi = p Af + p Bf 2. m A v Ai + m B v Bi = m A v Af + m B v Bf 3. Substitute numbers in and solve!

20. Conservation of Momentum in a 1-Dimensional Inelastic Collision p i = p f 1. p Ai + p Bi = p f 2. m A v Ai + m B v Bi = (m A + m B )v f 3. Substitute numbers in and solve!

21. Explosions Momentum is conserved in Explosions Before the Explosion, the initial momentum is zero (p i = 0) Therefore, the momentum after the explosion must be zero (p f = 0) The momenta must be equal and opposite after the explosion

22. Conservation of Momentum in Two Dimensions The conservation of momentum applies to all closed isolated systems in one, two and three dimensions Two Dimensional Conservation of Momentum Problems can be done with Vector Addition

23. Conservation of Momentum in Two Dimensions In doing two dimensional conservation of momentum problems; 1. Draw your momentum vectors before and after the collision 2. Treat p i = p f as a vector equation. Make a vector diagram (triangle) and solve for the momenta with vector addition 3. Solve for velocities after you find the momenta

24. Conservation of Angular Momentum Angular Momentum is also conserved for Rotating Objects. L i = L f I i  i = I f  f If the Rotational Inertia Changes then the Angular Velocity must change to keep the Angular Momentum (L) constant.