High mass objects can have low momentum when they have low velocities; low mass objects can have high momentum when they have high velocities. The more momentum an object has, the harder it is to stop.

Newton's second law of motion expressed in terms of momentum states that the rate of change in momentum of an object is equal to the net force applied to it.

The rate of change of momentum of a body is equal to the net force applied to it.

An example of momentum change and its magnitude: A ball is thrown at a wall, which stops it. The ball exerts a force on the wall equivalent to its momentum change. If you know the mass of the ball and the original velocity of the ball, you can calculate the momentum change. If you know the length of time it took to stop the ball, you can calculate the force on the ball.

Another example of momentum change and its magnitude: The ball is thrown at the wall and rebounds back toward the thrower. The change in momentum and thus the force is much greater in magnitude. The wall exerts a force to stop the ball but an additional force to give it momentum in the opposite direction. Remember, Dv = v_f - v_i. In the example described, assume v_i = -v_f = v, Dv = v_f - v_i= 2v, not zero.

Newton's Cradle demonstration

Collisions and Impulse In a collision of two ordinary objects, both objects are deformed. When the collision occurs, the force jumps from zero at the moment of contact to a very large quantity and back to zero. This occurs over a very brief instant of time. Impulse is useful when dealing with collisions because the forces involved in collisions are usually not constant. It is useful because it lets one calculate the average acceleration experienced by an object that is being acted upon by a non-constant force. The graph below represents force as a function of time during a collision. The curved area of the graph represents Dt, which is usually very small. The yellow area of the graph (the area under the curve) represents the product of force and Dt, or the impulse given to the object. The average force can be estimated from the graph and used with Dt to estimate the impulse. The same area under the curve, or the same impulse, could be obtained by a larger force acting over a shorter time interval or by a smaller force acting over a longer time interval.

According to Newton's second law, an unbalanced force causes a mass to accelerate. Restating Newton's second law in terms of momentum, an impulse causes the velocity of an object with mass to change, therefore causing a change in momentum

Impulse explains the operation of air bags, etc. An air bag increases the time that the change in momentum occurs over. The product of the force and Dt must equal the change in momentum. If Dt is increased, then the force must be decreased in order for their product to be constant.

Objects transfer their momentum in collisions. The total momentum before the collision is equal to the total momentum after the collision in a closed, isolated system. If one object loses momentum in a collision, then another object must gain that amount of momentum.

The law of conservation of energy previously studied is one of the conservation laws of physics. The law of conservation of momentum introduces another quantitity that is conserved in physics, linear momentum. Other quantities found to be conserved are angular momentum and electric charge.

There are two types of collisions:

• Elastic collision
  total momentum and total energy is conserved; elastic collisions only occur on the sub-atomic particle level

An Advanced look at Elastic Collisions  Assume two objects with mass m₁ and m₂ moving with intial speeds v₁ and v₂, respectively, collide head on in a perfectly elastic collision. Both momentum and kinetic energy are conserved in elastic collisions. Their final speeds can be written as v_1f and v_2f, respectively. If the objects have the same mass, they simply exchange velocities in a perfectly elastic collision.

The expression for conservation of linear momentum can be written as:
m₁v₁ + m₂v₂ = m₁v_1f + m₂v_2f
The expression for conservation of kinetic energy can be written as:
1/2m₁v₁² + 1/2m₂v₂² = 1/2m₁v_1f² + 1/2m₂v_2f²

Graphing calculator solution for elastic collisions

Inelastic collision

total momentum is conserved; real-life collisions are all inelastic.

An Advanced Look at Inelastic Collisions  If two objects stick together after the collision, it is said to be a totally inelastic collision. In inelastic collisions, some of the kinetic energy is converted into other types of energy such as thermal energy or potential energy.The sum of the final kinetic energies is always less than the sum of the initial kinetic energies in an inelastic collision.

Elastic and Inelastic Collisions Interactive Demonstration

Newton's Cradle

One-Dimensional Collision Applets

Interactive Two-Dimensional Collision Applet

Interactive linear collision

Interactive Conservation of Linear Momentum Applet

Method of working linear momentum problems:

1. Find the initial momentum of each object. Remember that momentum is a vector quantity--object's velocities are positive or negative. The total initial momentum is the sum of all the object's initial momentum.
2. Find the final momentum of each object. Remember that momentum is a vector quantity--object's velocities are positive or negative. The total final momentum is the sum of all the object's initial momentum
3. Set the total initial momentum equal to the total final momentum.

Be careful when working conservation of momentum problems! The problems can be algebraically correct, but not be correct according to the law of conservation of physics. Your physics must be correct--not just your algebra!

Method of working two-dimensional momentum problems:

Interactive 2-D collisions

1. When the velocity of the object is at an angle, resolve this resultant velocity into its x- and its y-components. Remember to assign a positive or a negative sign to the components' velocities.
2. Conserve the momentum in the x-direction. Find the initial momentum of each object in the x-direction. In other words, only consider the x-component of its velocity. Add the initial momentum of each object to find the total initial momentum. Find the final momentum of each object in the x-direction. In other words, only consider the x-component of its velocity. Add the final momentum of each object to find the total final momentum. Set the total initial momentum in the x-direction equal to the total final momentum in the x-direction.
3. Conserve the momentum in the y-direction. Find the initial momentum of each object in the y-direction. In other words, only consider the y-component of its velocity. Add the initial momentum of each object to find the total initial momentum. Find the final momentum of each object in the y-direction. In other words, only consider the y-component of its velocity. Add the final momentum of each object to find the total final momentum. Set the total initial momentum in the y-direction equal to the total final momentum in the y-direction.
4. Step two yields the x-component of the desired velocity. Step three yields the y-component of the desired velocity. Use vector addition of the two components to find the resultant velocity of the object.

AP Multiple Choice Questions for Momentum

1. Be prepared to perform simple calculations for linear momentum problems.
2. Remember signs for your velocities! For example, remember sign conventions for problems in which a ball rebounds from a bat, etc. The sign of its incoming velocity is opposite that of its outgoing velocity. This does not give you a net momentum change of zero!
3. Remember that momentum is conserved (the same) in any collision.
4. Be able to identify an elastic collision (objects rebound, do not stick together).
5. Remember that momentum is conserved, but kinetic energy is not, in inelastic collisions (which represent everything in real life).
6. Be able to recognize when momentum is changing from a graph. Momentum changes when speed changes.
7. Be able to predict the speeds and directions of objects after a collision.

AP Free Response Choice Questions for Momentum

1. Be able to perform calculations for ballistic pendulum problems. These are problems where an object is suspended in such a way that it can swing upward. It is hit with a moving object that becomes imbedded in the suspended object. This collision is the conservation of momentum part of the problem. You can use conservation of momentum to predict the speed of the suspended object/moving object (sometimes you work this problem "backwards" and use conservation of momentum to predict the speed of the moving object). The suspended object/moving object now has kinetic energy. It swings upward a height h, converting its kinetic energy into potential energy. You can use conservation of energy to predict how high it will swing upward.
2. Remember that energy lost in an inelastic collision usually results in a thermal energy gain (because of work done against friction).
3. Momentum is usually combined with kinetic energy calculations in free response questions.
4. Be able to perform calculations for objects attached to a spring. A moving object collides with the object on a spring and becomes stuck to it. The spring is compressed a distance d before it comes to rest. You can use conservation of momentum to predict the initial speed of the moving object/object attached to a spring. Its kinetic energy is now converted into elastic potential energy of the spring. Another variation -- collisions like this usually set the object into SHM, so they can ask you about the period of oscillation of the objects on the spring.
5. Use a graph or use data to estimate the average force acting on an object during a collision. Use this to calculate the impulse given to the object and its average acceleration. Remember, impulse better describes a collision where the force is variable.
6. You may be asked to work an inelastic collision problem, writing expressions for velocity. Then, they will ask you to do the same problem for an elastic collision!
7. Another calculation that is given that combines momentum and motion involves collisions occurring on rough surfaces. Remember that the fricitonal force serves as the unbalanced force that eventually causes the object to stop. You can predict how far it will go before it stops using the work/energy theorem or calculation the frictional force and using acceleration formulas and second law.
8. Not very common, but they have asked you to write velocity expressions before and after the collision for momentum in two dimension problems.

Momentum AP Objectives

Momentum Sample Problems

Linear Momentum and Impulse Homework

Momentum in Two Dimensions Homework

AP Momentum problems