Recoil and Collisions

8.01W07D1

Today’s Reading Assignment: W07D1

Young and Freedman: 8.3-8.4

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,

system surroundings 0Δ +Δ =p pr r

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: 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?

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

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

8

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 m1 is standing on a cart of mass m2 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 m3 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 m1 slides down a circular track of radius R cut into a large block of mass m2 as shown in the figure below. The large block rests on a , 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.

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Collisions

Collisions

Any interaction between (usually two) objects which occurs for short time intervals Δt 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:

K

0sys =K f

sys

1

2m

1v

1,02 +

12m2v2,0

2 +⋅⋅⋅=12m1v1, f

2 +12m2v2, f

2 +⋅⋅⋅

K

0sys > K f

sys

K

0sys < K f

sys

Demo: Ball Bearing and Glass B60

http://tsgphysics.mit.edu/front/index.php?page=demo.php?letnum=B%2060&show=0

Drop a variety of balls and let students guess order of elasticity.

Concept Question: Inelastic Collision

Cart A is at rest. An identical cart B, moving to the right, collides with cart A. They stick together. After the collision, which of the following is true?

1. Carts A and B are both at rest.

2. Carts A and B move to the right with a speed greater than cart B's original speed.

3. Carts A and B move to the right with a speed less than cart B's original speed.

4. Cart B stops and cart A moves to the right with speed equal to the original speed of cart B.

Concept Question: Inelastic Collision

A cart moving at speed v collides with an identical stationary cart on an airtrack, and the two stick together after the collision. What is their speed after colliding?

1. v

2. 1/2 v

3. zero

4. 2/3 v

5. 1/3 v

6. None of the above.

Concept Question: Elastic Collision

Cart A is at rest. An identical cart B, moving to the right, collides elastically with cart A. After the collision, which of the following is true?

1. Carts A and B are both at rest.

2. Cart B stops and cart A moves to the right with speed equal to the original speed of cart B.

3. Cart A remains at rest and cart B bounces back with speed equal to its original speed.

4. Cart A moves to the right with a speed slightly less than the original speed of cart B and cart B moves to the right with a very small speed.

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 M1. The ball on the bottom has a mass M2. 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 M2 >> M1.

M2>>M1

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 m1, is moving with an initial given velocity v1,0. The second ‘target’ particle is of mass m2 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 v1,f. Particle two moves off at an angle 2,f with a final velocity v2,fThe momentum diagram representing this collision is sown below.

Table Problem: Elastic Collision in 2-d

In the laboratory reference frame, an “incident” particle with mass m1, is

moving with given initial speed v1,i. The second “target” particle is of

mass m2 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 v1,f. Particle two moves off at an angle

2,f with final speed v2,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:

p

y,0sys =py, f

sys

p

x ,isys =px, f

sys

m

1v

1,i=m1v1, f cos1, f +m2v2, f cos2, f

0 =m1v1, f sin1, f −m2v2, f sin2, f

1

2m

1v

1,i2 =

12m1v1, f

2 +12m2v2, f

2

Strategy:• Three unknowns: v1,f , v2,f, and 2,f

• First squaring then adding the momentum equations and equations and solve for v2,f in terms of v1,f.

• Substitute expression for v2,f kinetic energy equation and solve quadratic equation for v1,f

• Use result for v1,f to solve expression for v2,f

• Divide momentum equations to obtain expression for 2,f

Next Reading Assignment: W07D2

Young and Freedman: 8.3-8.4

Experiment 4: Momentum and Collisions