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Physics 101 April 11, 2017 Lecture 21 Momentum (Ch. 9) Conservation of Momentum (Ch. 9) Collisions (Ch. 9) Elastic vs. Inelastic Collisions (Ch. 9) Momentum • Momentum, for a particle of mass , is defined as: and velocity – Momentum is a vector. – The SI unit of momentum is: • For a system consisting of momentum is: particles, the – If all particles have the same mass and velocity: M. Afshar 2 Momentum Conservation • Principle of momentum conservation: In the absence of external interactions, the total momentum of a system is constant in time. – “Absence of external interactions” means that the net external force is zero: . – Conservation of momentum requires that the net external force be zero; internal forces are irrelevant. M. Afshar 3 Momentum Conservation (cont.) • Momentum conservation can be expressed concisely as: If , then . • Other forms of momentum conservation: If , then . If , then . • In discussing momentum, we say that a system is “isolated” when . M. Afshar 4 How is Momentum Useful? • Momentum is similar to energy. – Momentum conservation relates initial momentum to final momentum. – Momentum conservation helps you calculate the final velocity when the initial velocity is known. • Four steps in using momentum conservation: 1. Choose your system carefully! It must be isolated. 2. Choose the initial time to be when all velocities are known. This will help you calculate initial momenta. 3. Use to solve for the final momentum. 4. Use to solve for the final velocity. M. Afshar 5 Practice Problem A canon resting on frictionless ice fires a projectile parallel to the surface. The canon recoils backward while the projectile moves forward. The canon and the projectile have masses and . If the projectile has speed , find the speed of the cannon immediately after it has fired. M. Afshar 6 Collisions • An important class of problems in physics involves collisions between two objects. • We will model collisions as follows: – Two objects are assumed to be isolated. – Masses and are known and fixed. – Initial velocities and are known. – We wish to find final velocities and . M. Afshar 7 Solving the Collision Problem • There are two internal forces: and . – These two forces are unknown and complicated. • Since the internal forces are unknown, we cannot calculate the acceleration of each object. – So we cannot use kinematics. • Since the internal forces are unknown, we cannot calculate the work they perform. – So we cannot use energy conservation. • Momentum conservation does not depend on internal forces. – So we can use momentum conservation. M. Afshar 8 Internal Forces • Why are internal forces irrelevant? – Short answer: Due to Newton’s third law, internal forces cancel each other, and thus cannot change the system’s momentum. – Long answer: , Since , Since , Since , Since M. Afshar 9 Practice Problem Two isolated particles and collide with initial speeds and . After collision rebounds with speed . a) What is the final velocity (speed and direction) of ? b) Is the total kinetic energy of the system conserved? a) b) No. (to the right) , . M. Afshar 10 Elastic vs. Inelastic Collisions • When two “isolated” bodies collide, momentum is always conserved. – Recall: If , then . • Is kinetic energy conserved? Not always! – Conserved for elastic collisions – kinetic energy remains as kinetic energy. • Example: Atomic collisions. – Not conserved for inelastic collisions – kinetic energy is transformed into other forms of energy. • Example: Car collisions. M. Afshar 11 Practice Problem Two isolated particles and collide with initial speeds and . The collision is elastic. What is the velocity (speed and direction) of each particle after the collision? Cons. of : Cons. of : , M. Afshar 12