General Physics I Momentum

Linear Momentum: Definition: For a single particle, the momentum p is defined as: p = mv (p is a vector since v is a vector). So px = mvx etc. Units of linear momentum are kg m/s.

Momentum Conservation The concept of momentum conservation is one of the most fundamental principles in physics. This is a component (vector) equation. Momentum along a certain direction is conserved when there are no external forces acting in this direction. Recall that force causes a chance in velocity and as momentum is defined by velocity….well, you see! You will see that we often have momentum conservation even when energy is not conserved.

1-D Conservation of Linear Momentum What this means is that the momentum “before” and the momentum “after” must be equal to each other. We will primarily deal with conservation of momentum with respect to 1-D linear collision situations For a 2-body system this is: m1v1i + m2v2i = m1v1f + m2v2f This is a component vector equation since velocity is a vector! So don’t forget that this means that the velocity could be positive or negative

Comment on Energy Conservation We have seen that the energy of a system is not necessarily conserved. Energy can be lost: Heat (bomb) Bending of metal (crashing cars) In terms of collisions, kinetic energy is not conserved since work is done during the collision! We talk about kinetic energy since, in the context of momentum, the object(s) has velocity. We will also deal with conservation of kinetic energy with respect to 1-D linear collision situations For a 2-body system this is: 1/2 m1v21i + 1/2 m2v22i = 1/2 m1v21f + 1/2 m2v22f

Elastic vs. Inelastic Collisions A collision is said to be elastic when kinetic energy as well as momentum is conserved before and after the collision. Kbefore = Kafter Carts colliding with a spring in between, billiard balls, etc. vi A collision is said to be inelastic when kinetic energy is not conserved before and after the collision, but momentum is conserved. Kbefore  Kafter Car crashes, collisions where objects stick together, etc.

Inelastic Collision in 1-D A block of mass M is initially at rest on a frictionless horizontal surface. A bullet of mass m is fired at the block with a muzzle velocity (speed) v. The bullet lodges in the block, and the block ends up with a speed V. In terms of m, M, and V : What is the initial energy of the system? What is the final energy of the system? What does momentum conservation tell us? Is kinetic energy conserved? x v V before after

Momentum Conservation Two balls of equal mass are thrown horizontally with the same initial velocity. They hit identical stationary boxes resting on a frictionless horizontal surface. The ball hitting box 1 bounces back with the same velocity it had initially, while the ball hitting box 2 gets stuck. Which box ends up moving faster? 1 2 V1 V2 (a) Box 1 (b) Box 2 (c) Same

Collisions A box sliding on a frictionless surface collides and sticks to a second identical box which is initially at rest. What is the ratio of initial to final kinetic energy of the system? (a) 1 (b) (c) 2

Inelastic collision in 2-D Consider a collision in 2-D (cars crashing at a slippery intersection...no friction). V v1 m1 + m2 m1 m2 v2 before after

Inelastic collision in 2-D... There are no net external forces acting. Use momentum conservation for both components. X: y: v1 V = (Vx,Vy) m1 + m2 m1 m2 v2

Explosion (inelastic un-collision) Before the explosion: M m1 m2 v1 v2 After the explosion: No external forces, so P is conserved. Initially: P = Finally: P =

The Crazy Bomb A bomb explodes into 3 identical pieces. Which of the following configurations of velocities is possible? m v V m v (1) (2) (a) 1 (b) 2 (c) Both

Elastic Collisions Elastic means that kinetic energy is conserved as well as momentum. This gives us more constraints We can solve more complicated problems!! Billiards (2-D collision) The colliding objects have separate motions after the collision as well as before. Let’s start with a simpler 1-D problem..... Initial Final

Elastic Collision in 1-D m2 m1 initial v1i v2i x m2 m1 final v2f v1f

Elastic Collision in 1-D m1 m2 before Conserve PX: m1v1i + m2v2i = m1v1f + m2v2f v1i v2i x after v1f v2f Conserve Kinetic Energy: 1/2 m1v21i + 1/2 m2v22i = 1/2 m1v21f + 1/2 m2v22f Suppose we know v1i and v2i We need to solve for v1f and v2f Should be no problem 2 equations & 2 unknowns! This relationship becomes: v1i – v2i = v2f – v1f

An example of 2-D elastic collisions: Billiards. If all we know is the initial velocity of the cue ball, we don’t have enough information to solve for the exact paths after the collision. But we can learn some useful things...

Billiards. Consider the case where one ball is initially at rest. pf pi vnet Pf F initial final The final direction of the red ball will depend on where the balls hit.

Billiards. The final directions are separated by 90o. pf pi vnet Pf F initial final

Billiards. So, we can sink the red ball without sinking the white ball.

Billiards. So, we can sink the red ball without sinking the white ball. However, we can also scratch. All we know is that the angle between the balls is 90o.

End of Momentum Lecture