Collisions Pg. 35 The previous lesson developed the idea of conservation of momentum, using the example of an “explosion”. This lesson examines conservation of momentum in collisions.

Objectives Physics terms Describe a perfectly inelastic collision. Apply conservation of momentum to collisions in one dimension. collision inelastic collision perfectly inelastic collision

Equations Conservation of momentum: Ask students to describe what the subscripts mean, such as “i2” (initial value for object number 2). Sometimes, initial velocity is given as v0, so make sure that this notation is also understood.

An interaction between two or more bodies in motion is a collision. What is a collision? An interaction between two or more bodies in motion is a collision. The definition of a collision.

What is a collision? Pool balls bouncing off of each other is one example. Collisions happen in the game of pool all the time. The collisions often take place in two dimensions, but the principles discussed in this lesson still apply.

What is a collision? Atoms repelling each other in a gas is a more important example. Collisions can involve particles interacting without making contact. This slide provides an example of this type of collision on the atomic scale.

What is conserved? What is conserved in a collision, and why? To answer this question, let’s look at what happens during a collision. This is the key question to be examined in this lesson and the next lesson.

What happens in a collision? What is happening in this collision between two balls? What might happen next? There are a couple possibilities. Ask students to describe a collision in terms of the physical quantities involved. Mass, velocity, momentum, and energy may all be mentioned. Ask students what kind of information we might want to know about a collision in a practical situation—such as in reconstructing what happened during a car accident. The outcome of this collision depends on the relative masses of the balls, and how elastic (bouncy) or inelastic (‘sticky”) they are.

What can’t happen in a collision? The red ball cannot move to the left, and the balls cannot pass through each other. Give students the opportunity to provide ideas of what cannot happen. If this is a 2D collision, for example, and the red ball moves up and right, then where can the green ball not move? (And so on.) Write down some answers, and return to them at the end of the lesson to evaluate which ones cannot happen (due to momentum conservation).

How do the velocities change? Here is one possible result of the collision. Examine the velocities shown in the figure. How do the velocities change? Velocity is often a key quantity in collisions when mass doesn’t change. Often we want to know how the velocities of objects change before and after a collision. Students might be confused by the three different velocities: one is the green ball before the collision, while the other two are the green ball and red ball after collision. Note that this is one possible outcome.

What forces are generated? What is true about these forces? Which of Newton’s laws applies here? Every collision involves forces. In the collision pictured above, the green ball exerts a force F on the red ball. By Newton’s third law, the red ball exerts an equal and opposite force on the green ball.

What forces are generated? What is the NET force on this system? system boundary The net force is zero. What is the system here? Is it open or closed? (Answer: closed.) Which context—closed or open—corresponds to conservation conditions? (Quantities are conserved in closed systems.)

What forces are generated? This is a closed system. No external forces act on the system, and the internal forces cancel each other. So what is conserved? The net force is zero. What is the system here? Is it open or closed? (Answer: closed.) Which context—closed or open—corresponds to conservation conditions? (Quantities are conserved in closed systems.)

Conservation laws The momentum before the collision equals the momentum after the collision. The energy before the collision equals the energy after the collision. But—the energy may be transformed. Momentum: Energy: Momentum always remains constant before and after a collision (for a closed system). In the above example, we consider the system to consist of the green and red balls. Energy is always conserved in collisions, but may be transformed between different types (e.g. mechanical, electrical, chemical, thermal, etc.), and therefore is difficult or impossible to keep track of. The next lesson treats the special case of perfectly elastic collisions, in which the kinetic energy is conserved.

Three types of collisions Perfectly inelastic collision: The objects stick together. Inelastic collision: These collisions are somewhat bouncy. Elastic collisions: These collisions are “perfectly” bouncy.

Three types of collisions Momentum is conserved in all three types of collisions. Momentum is always conserved – but we can’t apply conservation of momentum equations unless both colliding objects are included within the system boundary. For an object bouncing off the Earth, as shown, this is not practical.

Investigation Student ID login: 742 379 6973 In Investigation 11B you will experiment with inelastic collisions. http://www.essential-physics.com/TX/sbook Student ID login: 742  379  6973 -Click on the “investigation” icon -Scroll down & click on Investigation 11B- Collisions -Select the investigation icon -Complete simulation & answer ?’s

Investigation Perfectly inelastic collisions The interactive model simulates a perfectly inelastic collision between two balls. Select an initial velocity for the moving ball. Run the simulation for different combinations of masses for the red and green balls. For each combination, tabulate the masses and velocities. Examine the table for patterns in the data.

Fill in the table below using the simulation Fill in the table below using the simulation. You are getting the FINAL velocities for the balls once they have collided & stuck together. You need to CHOOSE your initial velocity for ball 1. The vocabulary terms that are relevant to this lesson are shown. In a perfectly inelastic collision the two objects stick together after impact.

Investigation Answer the following questions using the data collected in your table on the previous slide Describe the velocities before and after the collision when masses are equal. Describe the velocities when the red target ball has more mass. Describe the velocities when the green ball has more mass.

Conservation of momentum In a closed system, the momentum BEFORE the collision equals the momentum AFTER the collision:

Conservation of momentum

Conservation of momentum A green ball mass m1 collides with a red ball of mass m2: m1 m2 pi = pf m1 vi1 + m2 vi2 = m1 vf1 + m2 vf2

Conservation of momentum In this collision the red ball is initially at rest. vi1 m1 m2 pi = pf m1 vi1 + m2 vi2 = m1 vf1 + m2 vf2 Which term can we get rid of in this equation & why? Ask: “For the collision pictured on this slide, which term equals zero?”

Conservation of momentum vi1 So now what we have left is: m1 m2 pi = pf m1 vi1 = m1 vf1 + m2 vf2

Conservation of momentum The balls stick together after this collision. How do we show that in this equation? vi1 m1 m2 pi = pf m1 vi1 = m1 vf1 + m2 vf2

Conservation of momentum vi1 So FINALLY we have: m1 m2 pi = pf m1 vi1 = (m1 + m2) vf vf

Solving the problem A green ball has a mass of 5.0 kg and a velocity of 10 m/s. A red ball has a mass of 15 kg and is initially at rest. The balls collide and stick together. What is their resulting velocity? Before you plug in numbers, can you make any predictions about this velocity?

Solving the problem This is the end equation we found previously: Since we want the final velocity, how do we get the Vf by itself? Help the students realize that the final velocity should be less than 10 m/s. If the mass goes up, the velocity should go down.

Solving the problem Point out that the moving mass is now four times as big, with only one fourth the velocity.

Assessment Define a perfectly inelastic collision.

Assessment Define a perfectly inelastic collision. In a perfectly (or totally) inelastic collision, the two objects stick together after the collision. There is only one final, shared velocity.

Assessment A 10 kg puck moving with a velocity of +3.0 m/s strikes a second stationary 10 kg puck. The pucks stick together after the collision. What is their combined velocity after the collision?

Assessment The moving mass doubles, so the velocity is half as big. A 10 kg puck moving with a velocity of +3.0 m/s strikes a second stationary 10 kg puck. The pucks stick together after the collision. What is their combined velocity after the collision? The moving mass doubles, so the velocity is half as big.

Assessment A 10 kg puck moving with a velocity of +3.0 m/s strikes a second stationary 10 kg puck. The pucks stick together after the collision. What was the kinetic energy of the system before the collision? After the collision?

Assessment A 10 kg puck moving with a velocity of +3.0 m/s strikes a second stationary 10 kg puck. The pucks stick together after the collision. What was the kinetic energy of the system before the collision? After the collision? Was kinetic energy conserved? Point out to students that part of the usefulness of the law of conservation of momentum is that it applies to ANY closed system - even in situations where the mechanical energy is not conserved because the system is not ideal.