Recoil and Collisions 8.01 W07D1 Associated Reading Assignment: Young and Freedman:

Announcements Math Review Night this Week Tuesday 9-11 pm in Test Two Next Week Thursday Oct 27 7:30-9:30 Rooms TBA Pset 6 Due Tuesday at 9 pm Next Reading Assignment W07D2: Young and Freedman:

Conservation of Momentum: System and Surroundings: For a fixed choice of system, we can consider the rest of the universe as the surroundings. Then, by considering the system and surroundings as a new larger system, all the forces are internal and so change in momentum of the original system and its surroundings is zero,

Concept Question: Choice of System Drop a stone from the top of a high cliff. Consider the earth and the stone as a system. As the stone falls, the momentum of the system 1. increases in the downward direction. 2. decreases in the downward direction. 3. stays the same. 4. not enough information to decide.

Concept Question: Choice of System Answer 3: The system is approximately isolated with no external forces acting on the system so the momentum stays the same. (We are ignoring the effects of the sun and moon), The forces between the earth and the stone are internal forces and hence cancel in pairs.

Concept Question: Jumping on Earth Consider yourself and the Earth as one system. Now jump up. Does the momentum of the system 1.Increase in the downward direction as you rise? 2.Increase in the downward direction as you fall? 3.Stay the same? 4.Dissipate because of friction?

Concept Question: Jumping on Earth Answer 3: No external forces are acting on the system so the momentum is unchanged.

Recoil

Suppose you are on a cart, initially at rest on a track with very little friction. You throw balls at a partition that is rigidly mounted on the cart. If the balls bounce straight back as shown in the figure, is the cart put in motion? 1. Yes, it moves to the right. 2. Yes, it moves to the left. 3. No, it remains in place. Concept Question: Recoil

Answer: 2. Because there are no horizontal external forces acting on the system, the momentum of the cart, person and balls must be constant. All the balls bounce back to the right, then in order to keep the momentum constant, the cart must move forward. Concept Question: Recoil

Strategy: Momentum of a System 1. Choose system 2. Identify initial and final states 3. Identify any external forces in order to determine whether any component of the momentum of the system is constant or not 11

Problem Solving Strategies: Momentum Flow Diagram Identify the objects that comprise the system Identify your choice if reference frame with an appropriate choice of positive directions and unit vectors Identify your initial and final states of the system Construct a momentum flow diagram as follow: Draw two pictures; one for the initial state and the other for the final state. In each picture: choose symbols for the mass and velocity of each object in your system, for both the initial and final states. Draw an arrow representing the momentum. (Decide whether you are using components or magnitudes for your velocity symbols.)

A person of mass m 1 is standing on a cart of mass m 2 that is on ice. Assume that the contact between the cart’s wheels and the ice is frictionless. The person throws a ball of mass m 3 in the horizontal direction (as determined by the person in the cart). The ball is thrown with a speed u with respect to the cart. a)What is the final velocity of the ball as seen by an observer fixed to the ground? b)What is the final velocity of the cart as seen by an observer fixed to the ground? Table Problem: Recoil

Momentum Flow Diagram: Recoil

Table Problem: Sliding on Slipping Block A small cube of mass m 1 slides down a circular track of radius R cut into a large block of mass m 2 as shown in the figure below. The large block rests on a frictionless table, and both blocks move without friction. The blocks are initially at rest, and the cube starts from the top of the path. Find the velocity of the cube as it leaves the block. 15

Collisions

Any interaction between (usually two) objects which occurs for short time intervals  t when forces of interaction dominate over external forces. Of classical objects like collisions of motor vehicles. Of subatomic particles – collisions allow study force law. Sports, medical injuries, projectiles, etc.

Collision Theory: Energy Types of Collisions Elastic: Inelastic: Completely Inelastic: Only one body emerges. Superelastic:

Demo: Ball Bearing and Glass B60 demo.php?letnum=B%2060&show=0 Drop a variety of balls and let students guess order of elasticity.

Concept Question: Inelastic Collision Cart 2 is at rest. An identical cart 1, moving to the right, collides with cart 2. They stick together. After the collision, which of the following is true? 1.Carts 1 and 2 are both at rest. 2.Carts 1 and 2 move to the right with a speed greater than cart 1's original speed. 3.Carts 1 and 2 move to the right with a speed less than cart B's original speed. 4.Cart 1 stops and cart 2 moves to the right with speed equal to the original speed of cart 1.

Concept Question: Inelastic Collision Answer 3: From conservation of momentum, So Thus they move away with speed less one half the original speed of cart 1.

Concept Question: Elastic Collision Cart 2 is at rest. An identical cart 1, moving to the right, collides elastically with cart 2. After the collision, which of the following is true? 1.Carts 1 and 2 are both at rest. 2.Cart 1 stops and cart 2 moves to the right with speed equal to the original speed of cart 1. 3.Cart 2 remains at rest and cart 1 bounces back with speed equal to its original speed. 4.Cart 2 moves to the right with a speed slightly less than the original speed of cart 1 and cart 1 moves to the right with a very small speed.

Concept Question: Elastic Collision Answer: 2. Since there are no external forces, the momentum is constant and therefore The collision is elastic so conservation of energy implies Comparing equations There are two possibilities: Cart 1 stops and hence cart 2 moves with the initial speed of cart 1. The other possibility that Cart 2 is at rest just reproduces the initial conditions.

Demo and Worked Example: Two Ball Bounce Two superballs are dropped from a height h above the ground. The ball on top has a mass M 1. The ball on the bottom has a mass M 2. Assume that the lower ball collides elastically with the ground. Then as the lower ball starts to move upward, it collides elastically with the upper ball that is still moving downwards. How high will the upper ball rebound in the air? Assume that M 2 >> M 1. M 2 >>M 1

Table Problem: Three Ball Bounce Three balls having the masses shown are dropped from a height h above the ground. Assume all the subsequent collisions are elastic. What is the final height attained by the lightest ball?

Mini-Experiment: Astro-Blaster

Two Dimensional Collisions

Two Dimensional Collisions: Momentum Flow Diagram Consider a collision between two particles. In the laboratory reference frame, the ‘incident’ particle with mass m 1, is moving with an initial given velocity v 1,0. The second ‘target’ particle is of mass m 2 and at rest. After the collision, the first particle moves off at an angle  1,f with respect to the initial direction of motion of the incident particle with a final velocity v 1,f. Particle two moves off at an angle  2,f with a final velocity v 2,f The momentum diagram representing this collision is sown below.

Table Problem: Elastic Collision 2-d In the laboratory reference frame, an “incident” particle with mass m 1, is moving with given initial speed v 1,i. The second “target” particle is of mass m 2 and at rest. After an elastic collision, the first particle moves off at a given angle  1,f with respect to the initial direction of motion of the incident particle with final speed v 1,f. Particle two moves off at an angle  2,f with final speed v 2,f. Find the equations that represent conservation of momentum and energy. Assume no external forces.

Momentum and Energy Conservation No external forces are acting on the system: Collision is elastic:

Strategy: Three unknowns: v 1,f, v 2,f, and  2,f First squaring then adding the momentum equations and equations and solve for v 2,f in terms of v 1,f. Substitute expression for v 2,f kinetic energy equation and solve quadratic equation for v 1,f Use result for v 1,f to solve expression for v 2,f Divide momentum equations to obtain expression for  2,f