Conservation Laws Conservation of Momentum I

Total Momentum In momentum problems there is only one equation for momentum, but there will often be more than one mass. Finding the total momentum involves adding up the momentum of every mass in the problem. Since there is only one type of momentum this looks easier, and often is. What makes momentum difficult is that momentum and velocity are both vectors, and their directions are extremely important.

Example 1 A 4 kg mass is moving at 3 m/s to the right. A second mass of 2 kg is moving at 4 m/s to the left. What is their total momentum? The first mass you see will be m1 . If directions are given in the problem (as they are above) use them. If no directions are specified then make v1 positive. Then set the sign on v2 based on v1 . If m2 is moving in the same direction as m1 give v2 the matching sign, setting it as positive. If m2 is moving in the opposite direction compared to m1 , then give v2 the opposite sign by setting it as negative.

Conservation of Momentum Momentum cannot be created or destroyed. In our class linear momentum will always conserved. There are harder problems involving friction, etc. where momentum is lost, just as energy was lost in problems involving nonconservative forces. However, we won’t be responsible for these in this course. Momentum cannot change form, since there is only one kind of momentum. However, it can be passed from object to object. Momentum is the Physics of collisions. When objects collide with one another it is a conservation of momentum problem.

Elastic Collisions Collisions where the objects bounce apart without losing any energy. If two colliding masses touch each other they vibrate, which generates heat. This heat energy would be lost. In an elastic collision no energy is lost. This means that elastic collisions are perfect. The most perfect collisions would involve objects that do not touch each other during the collision. What kinds of collisions would do this? Proton to proton or electron to electron, 2nd semester Magnets with like poles, 2nd semester Masses with a spring between them. Right now this is the one to worry about.

Example 2 A mass m1 = 4 kg moving at 2 m/s to the right collides elastically with a mass m2 = 4 kg that is at rest. After the collision mass m1 is at rest. Determine the speed of m2 . 4 kg 2 m/s 0 m/s In this problem motion to the right is positive (+x direction) Right was defined as positive, so a positive sign on the answer is actually the direction. Specifying +x is a bit more sophisticated than saying “right”.

Inelastic Collisions Collisions where energy is lost Momentum does not change, but energy does. In an inelastic collision energy is lost. The objects do touch each other during the collision. The collision causes the objects to vibrate and vibration generates thermal energy. The thermal energy leaks out of the system as heat. It passes to the environment.

Kinetic Energy Energy is the ability to do work. Example Water falling in a water fall has speed. The falling water can hit the paddles of a wheel turning it. This can be used to run a machine. Since the falling water has the ability to work it has energy. The energy of moving masses is kinetic energy.

What happens with energy in collisions? Momentum, is always conserved when no external forces act. Objects with velocity also have kinetic energy. In elastic collisions kinetic energy is conserved and Klost = 0 In inelastic collisions kinetic energy is not conserved and there is Klost

Example 3 A mass m1 = 2 kg moving at 3 m/s to the right collides inelastically with a mass m2 = 1 kg moving at 6 m/s to the left. After the collision mass m1 is moving at 2 m/s to the left. 2 kg 3 m/s 1 kg 6 m/s a. Determine the speed and direction of m2 .

Example 3 A mass m1 = 2 kg moving at 3 m/s to the right collides inelastically with a mass m2 = 1 kg moving at 6 m/s to the left. After the collision mass m1 is moving at 2 m/s to the left. 2 kg 3 m/s 1 kg 6 m/s b. Determine the kinetic energy lost in the collision.

Perfectly Inelastic Collisions Collisions where the masses stick together Start with the usual conservation equation, and modify it. If they stick together, then there is one combined mass with one combined speed at the end. If they stick together, then they definitely touch. There will be an energy loss.

Example 4 A mass m1 = 1 kg moving at 3 m/s to the right collides perfectly inelastically with a mass m2 = 4 kg moving at 2 m/s to the left. 1 kg 3 m/s 4 kg 2 m/s a. Determine the speed of the system after the collision.

Example 4 A mass m1 = 1 kg moving at 3 m/s to the right collides perfectly inelastically with a mass m2 = 4 kg moving at 2 m/s to the left. 1 kg 3 m/s 4 kg 2 m/s b. Determine the kinetic energy lost in the collision.

Explosion The opposite of a perfectly inelastic collision Start with the usual conservation equation, and modify it. In an explosion the mass starts together and then breaks into smaller pieces.

Example 5 ? m/s You start with a 4 kg mass. 4 kg What is it doing ? A mass 4 kg mass explodes into two pieces. A 1 kg piece moves to the right at 6 m/s. What is the speed and direction of the other piece? You start with a 4 kg mass. 4 kg What is it doing ? ? m/s 6 m/s They did not say. Make the simplest assumption. Standing Still v0 = 0 3 kg 1 kg Then it splits into a 1 kg mass and a 2nd mass. The 2nd mass must be 3 Kg .