1 Collision Theory 8.01 W08D1 Today’s Reading Assignment: MIT 8.01 Course Notes: Chapter 15 Collision Theory Sections 15.1-15.5 Announcements Exam 2 Week 8 Thursday Oct 24 7:30-9:30 See Announcements Page for Room Assignments Conflict Exam: Fri 8-10 am in 26-328 or 10-12 noon in 26-168 Email [email protected] for request to take conflict exam Problem Set 6 due Week 8 Tuesday at 9 pm in box outside 26-152 Exam 2 Reviews: Tuesday 7:30-9 pm in 26-152 Wednesday 9-10:30 pm in 26-152 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 32 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= 0Concept 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.3 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: K0sys= Kfsys 12m1v1,02+12m2v2,02+ ⋅ ⋅ ⋅ =12m1v1, f2+12m2v2, f2+ ⋅ ⋅ ⋅ K0sys> Kfsys K0sys< KfsysDemonstration: Elastic and Inelastic Collisions Carts on track4 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 Pendulum5 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 12m1v1,x , i2+12m2v2, x ,i2=12m1v1,x , f2+12m2v2, x , f2 m1(v1,x , i2− v1,x , f2) = m2(v2, x , f2− 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) Kisys= KfsysElastic 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,isys= px, fsys m1(v1,x,i− v1,x, f) = m2(v2, x , f− v2, x , i)6 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 .7 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. M1M2>> M1M3>> M2 Problem: Three Ball BounceThree balls having the masses shown are dropped from
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