Physics
Vandebilt Catholic High School
W. Dupre

Momentum - Notes

1. Momentum - a measure of how hard it is to stop a moving object; it is the product of the object's mass and its velocity; since velocity is a vector, momentum is a vector quantity; symbol is p and SI unit are kg m/sec. The direction of the momentum is the same as that of the velocity.

p= m v ;(where p is momentum, m is mass in kg, and v is velocity in m/s)

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.

SF = Dp / D t

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.

2. Impulse - a force exerted over a time interval; symbol is I and SI unit is N s

I = F t ; (where I is impulse, F is force, and t is time in seconds)

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. 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

I = F t = m Dv = D p; (where D stands for "change in")

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.

3. System - a term that describes a collection of objects

a) Closed system - mass is constant
b) Open system - mass is not constant
c) Isolated system - one in which no external force acts

4. Law of conservation of momentum

the momentum of a closed, isolated system is constant; the sum of the initial momentum of the objects is equal to the sum of the final momentum of the objects

Sp_i = S p_f;(where p_i is the initial momentum and p_f is the final momentum)

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 quantity that is conserved in physics, linear momentum. Other quantities found to be conserved are angular momentum and electric charge.

5. 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. In an elastic collision there is no permanent deformity in the objects involved
An Advanced look at Elastic Collisions Assume two objects with mass m₁ and m₂ moving with initial 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

Inelastic collision – only total momentum is conserved, kinetic energy is not; real-life collisions are all inelastic. In inelastic collisions there is some deformity that occurs.
An Advanced Look at Inelastic Collisions If two objects stick together permanently after the collision, it is said to be a perfectly 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

One-Dimensional Collision Applets

Interactive Two-Dimensional Collision Applet

 

6. Solving Law of Conservation of Momentum Problems

a) Linear momentum - objects collide in straight-line motion. The collision occurs in a line, or one dimension.

Method of working linear momentum problems:

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.
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
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!

        

           b) Momentum in two dimensions - Momentum is conserved in the x-direction and in the y-direction.

        

           Method of working two-dimensional momentum problems:

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.
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.
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.
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.