Collision Theory

8.01 W08D1

Today’s Reading Assignment:

MIT 8.01 Course Notes: Chapter 15 Collision Theory

Sections 15.1-15.5

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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

3

Conservation of Momentum: System

For a fixed choice of system, if there are no external forces acting on the system then the momentum of the system is constant.

Δpsystem = 0

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?

Collisions Any interaction between (usually two) objects which occurs for short time intervals when interaction forces dominate over external forces. Examples: Collisions of motor vehicles.

Collisions of subatomic particles – collisions allow study force law.

Collisions in sports: medical injuries, projectiles, etc.

Suppose you are on a cart, initially at rest on a frictionless track. You throw balls at a partition that is rigidly mounted on the cart. After the balls bounce straight back as shown in the figure, is the cart 1. moving to the right? 2.  moving to the left? 3.  at rest.

Concept Question: Collision

Collision Theory: Energy Types of Collisions Elastic:

Inelastic:

Completely Inelastic: Only one body emerges.

Superelastic:

K0

sys = K fsys

12

m1v1,02 +

12

m2v2,02 + ⋅ ⋅ ⋅ =

12

m1v1, f2 +

12

m2v2, f2 + ⋅ ⋅ ⋅

K0

sys > K fsys

K0

sys < K fsys

Demonstration: Elastic and Inelastic Collisions

Carts on track

Concept Question: Inelastic Collision

Cart 2 is at rest on a fricitionless track. 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 2's original speed.

4.  Cart 1 stops and cart 2 moves to the right with speed equal to the original speed of cart 1.

Table Problem: Totally Inelastic Collision

A car of mass m1 moving with speed v1,i collides with another car that has mass m2 and is initially at rest on a frictionless track. After the collision the cars stick together and move with speed vf. What is the ratio ΔK/Ki = (Kf - Ki)/Ki?

Demonstration: Ballistic Pendulum

Table Problem: Ballistic Pendulum

A simple way to measure the speed of a bullet is with a ballistic pendulum, which consists of a wooden block of mass m1 into which a bullet of mass m2 is shot. The block is suspended from two cables, each of length L. The impact of the bullet causes the block and embedded bullet to swing through a maximum angle φ. Find an expression for the initial speed of the bullet as a function of m1, m2, g, and φ.

Elastic Collision: Conservation of Energy

Two particles interact elastically with no external forces along direction of motion: energy equation

12

m1v1,x ,i2 +

12

m2v2,x ,i2 =

12

m1v1,x , f2 +

12

m2v2,x , f2

m1(v1,x ,i

2 − v1,x , f2 ) = m2 (v2,x , f

2 − v2,x ,i2 )

m1(v1,x ,i + v1,x , f )(v1,x ,i − v1,x , f ) = m2 (v2,x , f + v2,x ,i )(v2,x , f − v2,x ,i )

Ki

sys = K fsys

Elastic Collision: Conservation of Momentum

Two particles interact elastically with no external forces along direction of motion: momentum equation

m1v1,x ,i + m2v2,x ,i = m1v1,x , f + m2v2,x , f

px ,i

sys = px , fsys

m1(v1,x ,i − v1,x , f ) = m2 (v2,x , f − v2,x ,i )

Elastic Collision: Conservation of Momentum and

Energy Summary: Divide bottom by top: Summary:

m1(v1,x ,i + v1,x , f )(v1,x ,i − v1,x , f ) = m2 (v2,x , f + v2,x ,i )(v2,x , f − v2,x ,i )

m1(v1,x ,i − v1,x , f ) = m2(v2,x , f − v2,x ,i )

v1,x ,i + v1,x , f = v2,x , f + v2,x ,i

v1,x ,i − v2,x ,i = v2,x , f − v1,x , f

m1(v1,x ,i − v1,x , f ) = m2(v2,x , f − v2,x ,i )

Concept Q.: Elastic Collision

Cart 2 is at rest on a frictionless track. 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.

Table Problem: One Dimensional Elastic Collision:

Consider the elastic collision of two carts on a frictionless track; the incident cart 1 has mass m1 and moves with initial speed v1,i . The target cart 2 has mass m2 = 2 m1 and is initially at rest. Immediately after the collision, the incident cart has final speed v1,f and the target cart has final speed v2,f. Find the final velocities of the carts as a function of the initial speed v1,i .

Worked Example: Gravitational Slingshot

A spacecraft of mass m1 with a speed v1i , approaches Saturn which is moving in the opposite direction with speed vs. After interacting gravitationally with Saturn, the spacecraft swings around Saturn and heads off in the opposite direction it approached . The mass of Saturn is ms. Find the final speed, v1f , of the spacecraft after it is far enough away from Saturn to be nearly free of Saturn’s gravitational pull.

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.

M1

M2 >> M1

M3 >> M2

������Problem: Three Ball Bounce

Three balls having the masses shown are dropped from aheight h. Assume all the subsequent collsions are elastic.What is the final height attained by the lightest ball?

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?

M1

M2 >> M1

M3 >> M2

Group Problem: Three Ball Bounce

Three balls having the masses shown are dropped from aheight h. Assume all the subsequent collsions are elastic.What is the final height attained by the lightest ball?

Mini-Experiment: Astro-Blaster

Two Dimensional Collisions

Strategies: Conservation of Momentum

If system is isolated, write down the condition that momentum is constant in each direction

m1 vx( )1,0

+ m2 vx( )2,0+ ⋅ ⋅ ⋅ = m1 vx( )1, f

+ m2 vx( )2, f+ ⋅ ⋅ ⋅

( psys,0 )x = ( psys, f )x

m1 vy( )

1,0+ m2 vy( )

2,0+ ⋅ ⋅ ⋅ = m1 vy( )

1, f+ m2 vy( )

2, f+ ⋅ ⋅ ⋅

( psys,0 ) y = ( psys, f ) y

Table Problem: Elastic Collision 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 θ1,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 θ2,f with final speed v2,f. Find the equations that represent conservation of momentum and energy. Assume no external forces. You do not have to solve these equations for θ1,f , θ2,f , and v2,f.