Linear Momentum

The linear momentum p of an object of mass m with a velocity of v is It is a vector and points in the same direction as the velocity vector.

The momentum vector is an entirely different vector than the velocity vector. Care should be taken in comparing one to the other. It is safe to say that the momentum vector is in the same direction as the velocity vector as mentioned earlier. One can also say that the momentum vector is directly proportional to the velocity vector, i.e., the momentum vector doubles if the velocity vector doubles.

But momentum also depends on the mass. So changing the mass of an object will also change the momentum vector. Therefore to change momentum one must change the mass or velocity or both. Regardless of what changes, the momentum vector is always in the same direction as the velocity vector.

As long as there are no external forces acting on a system of particles, collisions between the particles will exhibit conservation of linear momentum. This means that the vector sum of the momenta before collision is equal to the vector sum of the momenta of the particles afterwards.

This is known as the conservation of linear momentum. It is an extremely important concept in physics. One key area that makes use of this conservation principle is collisions. This is what you are going to explore today.

Collision between two objects of the same mass. One mass is at rest. Collision between two objects. One not at rest initially has twice the mass. Collision between two objects. One at rest initially has twice the mass. Simple Examples of Head-On Collisions (Energy and Momentum are Both Conserved)

Collision between two objects of the same mass. One mass is at rest. Collision between two objects. One not at rest initially has twice the mass. Collision between two objects. One at rest initially has twice the mass. Simple Examples of Head-On Collisions (Totally Inelastic Collision, only Momentum Conserved)

Collision between two objects of the same mass. One mass is at rest. Example of Non-Head-On Collisions (Energy and Momentum are Both Conserved) If you do a vector addition of the two momenta after collision, you find that it is equal to the total momentum before collision.

Velocity Components in Projectile Motion (In the absence of air resistance.) Note that the horizontal component of the velocity remains the same if air resistance can be ignored.

The collision you will study will involve two objects of equal mass colliding in a non-head-on collision in a horizontal plane and then undergoing projectile motion after the collision. Since the horizontal component of velocity remains constant for a projectile in free fall (shown in previous slide), the horizontal part of the projectile motion can be used to represent the horizontal component of the momentum after collision.

Here is an example of what you are going to do in the exercise today. You will roll a ball down the curved ramp. This represents the velocity as the ball left the table because the horizontal velocity of a projectile remains constant in the absence of air resistance. x y z