Chapter 10 Collisions

Review Momentum: If Fext = 0, then momentum does not change For continuous momentum transfer (Rockets): 7/19/04

Rockets: Continuous Momentum Transfer 7/19/04

Momentum in a Collision In a collision, objects only exert forces on each other, so Fext=0. Total momentum is conserved 7/19/04

Impulse During a collision, the momentum on an object changes This change in momentum is called “Impulse” When objects A and B collide 7/19/04

Impulse Recall: In the limit of small Dt: (constant force) (changing force) 7/19/04

Impulse in a Collision Different collisions with the same total impulse: Blue Dp/Dt Large Momentum Red Dp/Dt Small Large F: p changes rapidly Small F: p changes slowly 7/19/04

Example: The Impulsive Spiderman Spiderman, who has a mass of 70 kg, jumps from a train 5 meters high moving at 20 m/s (about 40 mph). He lands standing up, taking Dt = 0.1 s to stop himself after making contact with the ground. How much force did his knees feel? 7/19/04

Example Treat as collision between Spiderman and the ground Get force from the impulse: Initial: p = mvtotal Final: p = 0 7/19/04

Example Need to find vy: If he wasn’t a superhero, he’d break his legs! 7/19/04

Example What if he rolls on landing for Dt = 2 sec? Much easier on the knees! 7/19/04

Cannon Recoil Cannon: mc=1134 kg Ball: mb=13.6 kg Ball shot at ~ speed of sound  vb = 340 m/s The cannon and ball are initially at rest: pball = mballvball = (13.6kg)(340 m/s) = 4620 kg m/s So, pcannon= -4620 kg m/s 7/19/04

A rope can easily handle this much force without Cannon Recoil T pc Cannon recoil stopped in ~2 s by ropes. What is the tension in the ropes? A rope can easily handle this much force without breaking 7/19/04

Momentum Conservation in Different Frames Simple 1D problem v -v m m PTOT = mv - mv =0 Stick together 2m v=0 7/19/04

Momentum Conservation in Different Frames Same 1D problem viewed from right hand block, or with right hand block at rest 2v m m PTOT = 2mv + 0 = 2mv v 2 m 7/19/04

Changes in Momentum Independent of Frame Case 1 Case 2 i f i f Left mv 2mv mv Right -mv mv PTf – PTi = 0 – 0 = 0 PTf – PTi = 2mv – 2mv = 0 7/19/04

Center of Momentum Frame There is always a frame of reference where PTOT=0. ‘Center of mass’ frame 7/19/04

A Limitation of Momentum vT vc V=30 MPH V=0 BOOM! Before After How do we determine the velocities? 7/19/04

A Limitation of Momentum ptruck pcar There are many possibilities Conservation of Momentum can’t tell them apart 7/19/04

Elastic Collisions Two equations: Momentum and kinetic energy are conserved Two equations: Good approximation for a lot of collisions, and exact for some Examples: Billiard Balls, superball on floor… 7/19/04

Elastic Collisions in One Dimension Before V1,i V2,i m1 m2 After V1,f V2,f Two conservation laws Momentum (Always) Energy (Elastic only - Mechanical Energy is conserved) 7/19/04

We now have two equations and two unknowns: A Unique Solution We now have two equations and two unknowns: Lots of Algebra 7/19/04

Limiting Cases How do we understand what types of motion these predict? Consider limiting case: m1 = m2 The two objects simply trade values of velocity! 7/19/04

Limiting Cases What if m1 >> m2? Semi truck hits a parked VW bug: Truck keeps going Bug bounces off with twice truck’s speed! 7/19/04

Demonstration m1>>m2 A Question: What Happens? Before: After:

The Slingshot Effect 9.6 km/s -10 km/s

Car-Truck Crash A 2000 kg car has a head-on collision with a 10,000 kg truck. They each are travelling at 10 m/s and they collide elastically (solid bumpers!). What are their final velocities? m1 v1i v2i m2 +x Choose positive x direction 7/19/04

Car-Truck Crash (continued) m1 v1i v2i m2 v1i = 10 m/s m1 = 2,000 kg v2i = -10 m/s m2 = 10,000 kg v2f = -3.33 m/s v1f = -23.3 m/s Truck slows down Car goes flying backwards! 7/19/04

Car-Truck Crash (continued) If the two vehicles are being driven by 60 kg PSU students, what are the impulses they feel? In truck: J = Dp = mDv = m(v2f - v2i) = 60(-3.33 – (-10)) = 400 kg m/s In car: J = Dp = mDv = m(v1f – v1i) = 60(-23.3 – (10)) = -2000 kg m/s 7/19/04

Car-Truck (question) Which would you rather be driving? Say collision lasts Δt = 0.2 seconds Force on student is given by F = Δp/Δt Student in truck feels 2,000 N (survivable) Student in car feels 10,000 N (not good) What if instead of a 2000 kg car, she was on a 500 kg motorcycle! 7/19/04

Example: 2-D Elastic Collision v1,i=(1 m/s)i+(2 m/s)j Two billiard balls collide elastically on a table. The initial velocity of the first ball is v1,i=(1 m/s)i+(2 m/s)j. The second ball is initially at rest. Both balls have the same mass. Determine the final velocity of both after the collision. 7/19/04

Inelastic Collisions Momentum is conserved (NOT Kinetic Energy) Completely Inelastic: Two objects stick together Examples: Spit wads, football player being tackled,… 7/19/04

Inelastic Collisions… http://www.baylortv.com/streaming/000026/300kbps_ref.mov 7/19/04

Car Crash m1=750 kg m2=1000 kg v1=20 m/s v2=30 m/s Two cars collide and stick together after the collision. What is the final velocity of the system? 7/19/04

Car Crash Using conservation of momentum: m1=750 kg m2=1000 kg v1=20 m/s v2=30 m/s Using conservation of momentum: 7/19/04

Basketball Cannon A ball projected from a cannon hits the trash can such that: It sticks into the trash can. It hits the trash can and bounces back. Will the velocity of the trash can be bigger for case 1, case 2, or exactly the same? 7/19/04

Basketball Cannon Consider an elastic collision: M m v vtrash=0 7/19/04

Basketball Cannon Consider a perfectly inelastic collision: M m v vtrash=0 Consider a perfectly inelastic collision: 7/19/04

Basketball Cannon Elastic: Inelastic: Elastic collision results in twice the velocity! 7/19/04