Law of Conservation of Momentum

If the resultant external force on a system is zero, then the vector sum of the momentums of the objects will remain constant.  ρ before =  ρ after

Sample problem A 75-kg man sits in the back of a 120-kg canoe that is at rest in a still pond. If the man begins to move forward in the canoe at 0.50 m/s relative to the shore, what happens to the canoe?

External versus internal forces External forces: forces coming from outside the system of particles whose momentum is being considered. –External forces change the momentum of the system. Internal forces: forces arising from interaction of particles within a system. –Internal forces cannot change momentum of the system.

Explosions When an object separates suddenly, as in an explosion, all forces are internal. Momentum is therefore conserved in an explosion. There is also an increase in kinetic energy in an explosion. This comes from a potential energy decrease due to chemical combustion.

Recoil Guns and cannons “recoil” when fired. This means the gun or cannon must move backward as it propels the projectile forward. The recoil is the result of action-reaction force pairs, and is entirely due to internal forces. As the gases from the gunpowder explosion expand, they push the projectile forwards and the gun or cannon backwards.

Sample problem Suppose a 5.0-kg projectile launcher shoots a 209 gram projectile at 350 m/s. What is the recoil velocity of the projectile launcher?

Sample Problem A firecracker is placed in a pumpkin which explodes into exactly two pieces. The first piece has a mass of 2.2 kg and flies due east at 26 m/s. The second chunk heads due west at 34 m/s. What was the initial mass of the pumpkin?

Collisions When two moving objects make contact with each other, they undergo a collision. Conservation of momentum is used to analyze all collisions. Newton’s Third Law is also useful. It tells us that the force exerted by body A on body B in a collision is equal and opposite to the force exerted on body B by body A.

Collision Types Elastic collisions –Also called “hard” collisions –No deformation occurs, no kinetic energy lost Inelastic collisions –Deformation occurs, kinetic energy is lost Perfectly Inelastic (stick together) –Objects stick together and become one object –Deformation occurs, kinetic energy is lost

(Perfectly) Inelastic Collisions Simplest type of collisions. After the collision, there is only one velocity, since there is only one object. Kinetic energy is lost. Perfectly inelastic collisions are the reverse of explosions in which kinetic energy is gained!

Sample Problem An 80-kg roller skating grandma collides inelastically with a 40-kg kid. What is their velocity after the collision? How much kinetic energy is lost?

Sample Problem A fish moving at 2 m/s swallows a stationary fish which is 1/3 its mass. What is the velocity of the big fish after dinner?

Sample problem A car with a mass of 950 kg and a speed of 16 m/s to the east approaches an intersection. A 1300-kg minivan traveling north at 21 m/s approaches the same intersection. The vehicles collide and stick together. What is the resulting velocity of the vehicles after the collision?

Elastic Collision In elastic collisions, there is no deformation of colliding objects, and no change in kinetic energy of the system.  p b =  p a (momentum conservation)  K b =  K a (kinetic energy conservation)

A couple tips Always start with conservation of momentum and only move on to conservation of kinetic energy if you don’t have enough information to solve the conservation of momentum equations

A couple tips When objects stick together, the collision must be inelastic, but if objects bounce off each other, it can be elastic OR inelastic. SO only use conservation of kinetic energy if the problem says that it is an elastic collision OR if the problem asks you if the collision is elastic and you use it to test and see if kinetic energy is conserved

Sample Problem A 500-g cart moving at 2.0 m/s on an air track strikes a 1,000-g cart at rest in an elastic collision. What are the resulting velocities of the two carts?

Example Two carts of equal mass move towards each other with identical speeds of 0.3 m/s. After colliding, the carts bounce off each other, each regaining 0.30 m/s of speed, but now moving in the opposite direction. Is this an elastic collision?

2D-Collisions Momentum in the x-direction is conserved. –  P x (before) =  P x (after) Momentum in the y-direction is conserved. –  P y (before) =  P y (after) Treat x and y coordinates independently. –Ignore x when calculating y –Ignore y when calculating x Let’s look at a simulation: –

Sample problem Calculate velocity of 8-kg ball after the collision. 3 m/s 2 kg 8 kg 0 m/s Before 2 m/s 2 kg 8 kg v After 50 o x y x y

Motion of the Center of Mass The center of mass of a system of objects obeys Newton’s second law. To find the center of mass –If the objects are of equal mass, the center of mass is directly between them –If the objects are not of equal mass, set up a proportion (the center of mass will be closer to the heavier mass)

Example A toy rocket is in projectile motion, so that it is on track to land 30 m from its launch point. While in the air, the rocket explodes into two identical pieces, one of which lands 35 m from its launch point. Where does the first piece land? 25 m from its launch point. Since the only external force acting on the rocket is gravity, the center of mass must stay in projectile motion, and must land 30 m from the launch point. The two pieces are of equal mass so if one is 5 m beyond the center of mass landing point, one must be 4 m short of that point.

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Ballistics Pendulum A ballistics pendulum was used in munitions factory to test the velocity of the bullets manufactured. A bullet is fired into a block of wood hanging on a string. The height to which the pendulum swings is measured and from this the initial velocity of the bullet can be measured.

Example A bullet with mass of 0.05 kg is fired into a ballistics pendulum with mass 3 kg. It swings to a height of 0.32 m. What is the velocity of the bullet?