Momentum and Impulse

What does it mean to have momentum?

Sports Example Coach: “We have all the momentum. Now we need to use that momentum and bury them in the third quarter.” What does the coach mean? His team is on the move and will take some effort to stop.

Momentum All objects in motion have momentum! What factors determine how much momentum an object has?

Linear Momentum Linear momentum (p) – product of an object’s mass and velocity. Momentum is a vector that points in the same direction as the velocity. Units: kg *m/s

Example #1: A freight train moves due north with a speed of 1. 4 m/s Example #1: A freight train moves due north with a speed of 1.4 m/s. The mass of the train is 4.5 x 105 kg. How fast would an 1800 kg automobile have to be moving due north to have the same momentum? ptrain = mtrainvtrain =(4.5 x 105 kg)(1.4 m/s) = 630,000 kg*m/s pauto = mautovauto = 350 m/s

Varying Forces Forces are not always constant. Many forces change as they are applied over a period of time. When a baseball hits a bat, the ball is in contact with the bat for a short time interval. The force reaches its maximum value towards the middle of the time interval. Average Force (F) t0 tf

Impulse For a baseball to be hit well, both the size of the force and the time of contact (Follow through!) are important. Impulse (J) – product of the average force F and the time interval Δt over which the force acts. Units: Newton * seconds (Ns) Impulse is a vector that points in the same direction as the average force.

Impulse-Momentum Theorem Remember Newton’s 2nd Law: And acceleration is defined as Substitute: Rearrange: Impulse-Momentum Theorem

Impulse-Momentum Theorem Another way to write the theorem is:

Example #2: A volleyball is spiked so that its incoming velocity of +4 Example #2: A volleyball is spiked so that its incoming velocity of +4.0 m/s is changed to an outgoing velocity of -21 m/s. The mass of the volleyball is 0.35 kg. What impulse does the player apply to the ball? v0 = +4.0 m/s vf = -21 m/s m = 0.35 kg

Why doesn’t the egg break?? Impulse Why doesn’t the egg break?? http://www.youtube.com/watch?v=7RSUjxiZnME

This can come from any combination of Force x time. Impulse An object with 100 units of momentum must experience 100 units of impulse to be brought to a stop . This can come from any combination of Force x time. As time Force

Conservation of Momentum closed system – system where nothing is lost and no net external forces act. Law of Conservation of Momentum – the total momentum of any closed system does not change; the momentum before an interaction equals the momentum after the interaction.

Collisions in 1-Dimension Inelastic Inelastic collision- a collision in which the objects stick together and move with one common velocity after colliding. 2 objects  1 object

Inelastic Collisions Example 3: Two cars collide and become entangled. If car #1 has a mass of 2000 kg and a velocity of 20 m/s to the right, and car #2 has a mass of 1500 kg and a velocity of 25 m/s to the left, find the velocity of the system after the collision. BEFORE Collision AFTER Collision 1 2 1 2

1 object  2 objects (or more) Explosions Explosion- process of one object splitting into 2 (or more) objects Recoil- kickback; momentum opposite a projectile 1 object  2 objects (or more)

Explosions Example 4: Starting from rest, two skaters push off each other on smooth level ice. Skater 1 has a mass of 88 kg and Skater 2 has a mass of 54 kg. Upon breaking apart, Skater 2 moves away with a velocity of 2.5m/s to the right. Find the recoil velocity of Skater 1. BEFORE Explosion AFTER Explosion 1 2 1 2

Collisions in 1-Dimension Elastic Elastic collision – a collision in which the colliding objects bounce off each other. 2 objects  stay 2 objects

Elastic Collisions BEFORE Collision AFTER Collision ? ? 1 2 1 2 This is the hardest type of problem since there are two pieces on both sides! Example #5: A ball of mass 0.250 kg and a velocity of 5.00 m/s to the right collides head-on with a ball of mass 0.800 kg that is initially at rest. No external forces act on the balls. If the collision is elastic, and the 0.800 kg ball leaves the collision with a speed of 2.38 m/s, what is the velocity of the other ball after the collision?

If there are 2 unknowns we need to use Conservation of Energy! Elastic Collisions BEFORE Collision AFTER Collision ? ? 1 2 1 2 If there are 2 unknowns we need to use Conservation of Energy!

Is Kinetic Energy Conserved? Elastic Collision: KE is conserved Inelastic Collision: KE is NOT conserved

Elastic Collisions with Energy Example: A ball of mass 0.250 kg and a velocity of 5.00 m/s to the right collides head-on with a ball of mass 0.800 kg that is initially at rest. No external forces act on the balls. If the collision is elastic, what are the velocities of the balls after the collision? v1f = -2.62 m/s v2f = +2.38 m/s

Elastic Collisions with Energy A cue ball of mass 0.165 kg is at rest on a frictionless pool table. The ball is hit dead center by a pool stick which applies an impulse to the ball causing it to roll at a velocity of 9.09 m/s. The ball then makes an elastic head-on collision with a second ball of mass 0.140 kg that is initially at rest. Find the velocities of each ball after colliding.

Remember Vectors??? Momentum is a vector, it follows all rules of vectors that we have previously learned. How do we solve for the Resultant vector? Break into Components Add x components, Add y components Pythagorean Theorem Trig Function Pythagorean Theorem Trig Function (tan) to find angle

2-D Collisions Inelastic Elastic

p0x = pfx p0y = pfy 2-D Collisions The x and y components of the momentum are conserved separately. p0x = pfx p0y = pfy

2-D Collisions: Inelastic Example 1: A 200 kg car going 2 m/s east collides with a 100 kg car moving 3 m/s north. If they lock bumpers, how fast and in what direction will they be going after the collision?

2-D Explosion A fireworks rocket is moving at a speed of 45.0 m/s. The rocket suddenly breaks into two pieces of equal mass, which fly off with velocities v1 and v2, as shown in the drawing. What is the magnitude of (a) v1 and (b) v2?

2-D Collisions: Elastic v01 = 0.900 m/s m1 = 0.150 kg vf1 50.0° Θ 35.0° v02 = 0.540 m/s m2 = 0.260 kg vf2 = 0.700 m/s

Find the missing values: v01 = 0.850 m/s m1 = 0.230 kg Find the missing values: vf2 and θ vf1 = 0.600 m/s 35.0° 15.0° Θ v02 = 0.430 m/s m2 = 0.190 kg vf2