### Momentum

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; it is a vector quantity; symbol is p and SI units are either N sec or kg m/sec.

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

Impulse
a force exerted over a time interval; symbol is J and SI units are the same as those for momentum

J = F t

where J is impulse, F is force, and t is time in seconds

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

J = F t = m Dv = D p

where D stands for "change in"

Angular momentum
the product of a rotating object's moment of inertia and its angular velocity; if there is no torque acting on an object, its angular momentum is constant

System
a term that describes a collection of objects

Closed system
mass is constant

Open system
mass is not constant

Isolated system
one in which no external force acts

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

Spi = S pf
where pi is the initial momentum and pf 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.

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
• Inelastic collision
total momentum is conserved; real-life collisions are all inelastic
Graphing calculator solution for elastic collisions

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

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!

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

Method of working two-dimensional momentum problems:

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 x-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.