Momentum and Impulse

1. Momentum and Impulse

2. Introduction to Momentum • Momentum is the “quantity of motion.” • Momentum is represented by “p”. • Momentum is the product of the mass and the velocity of an object or: p=mv • Momentum is a vector quantity, meaning it has both magnitude and direction. • Measured in units of kg m/s

3. Introduction to Impulse • Impulse is the product of the net force acting on a given body during a given time, or the change in momentum. Units = Ns = kg m/s • Impulse = f(Δt) = m(Δv) = Δp • “A large change in momentum occurs only when there is a large impulse. A large impulse, however, can result from either a large force acting over a short time, or a smaller force acting over a longer time.” F* t = F * t

4. Conservation of Momentum • The total momentum of a system of objects must remain constant unless outside forces act upon the system. • Momentum, like other physical quantities, cannot be destroyed, merely redistributed. • Momentum is distributed through impulse, or the change in momentum • Most studies of the conservation of momentum involve collisions.

5. For a collision occurring between object 1 and object 2 in a closed, isolated system, the total momentum of the two objects before the collision is equal to the total momentum of the two objects after the collision. That is, the momentum lost by object 1 is equal to the momentum gained by object 2. A closed isolated system is a set of objects which encounter no forces or influences other than those they exert on each other.

6. Types of Collisions • Collisions that Stick involve two or more objects that collide and stick together, forming an object with a mass equal to the sum of all of the other masses. • Collisions that Bounce involve two or more objects that collide and then separate. They could be going in the same direction after impact or in opposite directions.

7. Collisions that Stick • For Collisions that Stick, the end mass is the sum of all other masses, and velocity is a whole new number • Remember: initial momentum of the system = final momentum of the systesm. • m1v1+m2v2= (m1+m2) vf v1 v2 m1 m2 vf (m1 + m2)

8. Collisions That Bounce Apart • Remember: pinitial = pfinal • m1v1+m2v2 =m1v1’+m2v2’ • Usually mass remains the same after the collision but this does not always happen. Before v1 m1 m2 After v1’ v2’ m1 m2

9. Separations/Recoil • Sometimes objects start as one and separate into two parts moving with individual velocities. • Example: A bullet is fired from a rifle. The bullet leaves the rifle with a velocity of vb and as a result the rifle has a recoil velocity of vr. • Solution: Before the rifle is fired the momentum of the system is zero. So… 0 = mbvb’ + mrvr’ • Note: for total momentum to remain zero, the rifle’s velocity must be in the opposite direction to the bullet’s velocity.

10. Consider a collision between two objects - object 1 and object 2. For such a collision, the forces acting between the two objects are equal in magnitude and opposite in direction (Newton’s Third Law)

11. The forces act between the two objects for a given amount of time. • Regardless of how long the time is, it can be said that the time that the force acts upon object 1 is equal to the time that the force acts upon object 2.

12. Since the forces between the two objects are equal in magnitude and opposite in direction, and since the times for which these forces act are equal in magnitude, it follows that the impulses experienced by the two objects are also equal in magnitude and opposite in direction.

13. Since each object experiences equal and opposite impulses, it follows logically that they must also experience equal and opposite momentum changes.

14. In a collision, the momentum change of object 1 is equal and opposite to the momentum change of object 2. That is, the momentum lost by object 1 is equal to the momentum gained by object 2.

15. Consider a collision in football between a fullback and a linebacker during a goal-line stand. The fullback plunges across the goal line and collides in midair with linebacker. The linebacker and fullback hold each other and travel together after the collision. The fullback possesses a momentum of 100 kg*m/s, East before the collision and the linebacker possesses a momentum of 120 kg*m/s, West before the collision. The total momentum of the system before the collision is 20 kg*m/s, West.

16. Therefore, the total momentum of the system after the collision must also be 20 kg*m/s, West. The fullback and the linebacker move together as a single unit after the collision with a combined momentum of 20 kg*m/s. Momentum is conserved in the collision.