02-01-2013, 03:40 PM
Linear Momentum
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INTRODUCTION
Assume your file cabinet is sitting on a cart with wheels. It is sitting in the middle of a room with a smooth floor. You want to move it against the wall. You give it a push. It takes off, and before you know it, it slams into the wall. It is hard to stop it, because it has linear momentum.
Linear momentum is a measure of an object's translational motion. The linear momentum p of an object is defined as the product of the object's mass m times its velocity v.
p = mv.
Linear momentum is a vector. Its direction is the direction of the velocity.
Conservation of momentum
Consider two interacting objects. If object 1 pushes on object 2 with a force F = 10 N for 2 s to the right, then the momentum of object 2 changes by 20 Ns = 20 kgm/s to the right. By Newton's third law object 2 pushes on object 1 with a force F = 10 N for 2 s to the left. The momentum of object 1 changes by 20 Ns = 20 kgm/s to the left. The total momentum of both objects does not change. For this reason we say that the total momentum of the objects is conserved.
Newton's third law implies that the total momentum of a system of interacting objects that are not acted on by outside forces is conserved.
The total momentum in the universe is conserved. The momentum of a single object, however, changes when a net force acts on the object for a finite time interval. Conversely, if no net force acts on an object, its momentum is constant. For a system of objects, a component of the momentum along a chosen direction is constant, if no net outside force with a component in this chosen direction acts on the system.
Collisions
In collisions between two isolated objects Newton's third law implies that momentum is always conserved. Collisions in which the kinetic energy is also conserved, i.e. in which the kinetic energy just after the collision equals the kinetic energy just before the collision, are called elastic collision. In these collisions no ordered energy is converted into thermal energy. Collisions in which the kinetic energy is not conserved, i.e. in which some ordered energy is converted into internal energy, are called inelastic collisions. If the two objects stick together after the collision and move with a common velocity vf, then the collision is said to be perfectly inelastic.
Balls hitting balls
Assume two balls of equal mass, made from the same material, approach each other head on. Both balls have the same speed v. They approach each other with relative speed 2v. As the balls collide, each ball exerts a force on the other. The forces are equal in magnitude but have opposite directions. The balls distort like spherical springs, and the same amount of energy is stored in each ball as elastic potential energy. It will be reconverted into kinetic energy. The force with which ball 1 pushes on ball 2 first decelerates ball 2 to a stop and then accelerates it into a direction opposite its initial velocity. The force with which ball 2 pushes on ball 1 decelerates ball one to a stop and then accelerates it into a direction opposite its initial velocity. We expect the two balls to fly apart with equal speeds in opposite directions. If the coefficient of restitution of the two balls is 1, then their speed will not have changed. The total momentum of the two balls is conserved.
Bats and baseballs
Assume a baseball hits a stationary bat. If the bat is nailed to the wall of a house and cannot move at all, then the ball will just rebound the same way it rebounds from a hard floor. If the bat is held in the hand of the batter, then the force the ball exerts on the bat will accelerate the bat backwards, and some collision energy will be transferred to the bat and will not appear as rebound energy of the ball.
If the batter and the bat are very heavy, they receive little of the collision energy, and the ball rebounds with outgoing speed equal to the coefficient of restitution times the incoming speed.
If the batter is swinging the bat forward as the ball hits it, the ball's outgoing speed will be much higher. Assume the bat and the ball each are moving with speed 100 km/h in opposite direction. Their relative speed is 200 km/h. A reference frame in which the bat is stationary is moving with 100 km/h speed with the bat. In this reference frame the ball approaches with 200 km/h and rebounds with speed v = coefficient of restitution times 200 km/h in the forward direction. But this reference frame is moving itself with speed 100 km/h in the forward direction. With respect to the ground the ball is therefore moving with speed vground = 100km/h + v = 100km/h + coefficient of restitution * 200 km/h.