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# MIT 8 01 - Collision Theory

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1Collision Theory8.01Week 09D2CollisionsAny 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.Momentum and External Forces 1. Select your system.2. Identify all forces acting on the system.3. If there is a non-zero total external force,4. or collision time is instantaneous implies momentum is constant,sys sys0.f=ppGGtotalext=F0GGsystotalext.ddt=pFGGProblem Solving Strategies: Momentum• Identify the objects that compose the system.• Identify your initial and final states of the system. Draw two momentum diagrams: one for the initial state and the other for the final state. • Choose symbols to identify each mass and velocity in the system. • Identify a set of positive directions and unit vectors for each state.• Decide whether you are using components or magnitudes for your velocity symbols.• Momentum is a vector!2Momentum AnalysisSince momentum is a vector quantity, identify the initial and final vector components of the total momentum.Initial Statex-comp: y-comp:Final Statex-comp: y-comp:sys011,022,0mm=+ +⋅⋅⋅pvvGGGsys11, 2 2,fffmm= + +⋅⋅⋅pvvGGG()()sys,0 1 21,0 2,0xx xpmv mv=+ +⋅⋅⋅() ()sys,0 1 21,0 2,0yy ypmv mv= + +⋅⋅⋅() ()sys,1 21, 2,xf x xffpmv mv=+ +⋅⋅⋅()()sys,1 21, 2,yf y yffpmv mv=+ +⋅⋅⋅Conservation of MomentumIf no external forces are acting on the system, write down the condition that momentum is constant in each direction.()()()()12 1 21,0 2,0 1, 2,xx x xffmv mv mv mv++⋅⋅⋅ = + +⋅⋅⋅sys sys,0 ,yyfpp=px,0sys= px, fsys()()()()12 1 21,0 2,0 1, 2,yy y yffmv mv mv mv++⋅⋅⋅ = + +⋅⋅⋅Planar Collision Theory: EnergyTypes of Collisions in Two Dimensions: Elastic: Inelastic: Completely Inelastic: Only one body emerges.Superelastic: K0sys= Kfsys 12m1v1,02+12m2v2,02+⋅⋅⋅=12m1v1, f2+12m2v2, f2+⋅⋅⋅ K0sys> Kfsys K0sys< KfsysConcept Question: Elastic collisionCart 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.3Concept Question: Inelastic collisionCart 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.Table Problem: totally inelastic collisionA car of mass mAmoving with speed vA,1collides with another car that has mass mBand is initially at rest. After the collision the cars stick together and move with speed v2. What is the change in mechanical energy due to the collision?Table Problem: elastic collisionConsider the elastic collision of two carts along a track; the incident cart A has mass mAand moves with initial speed vA,1. The target cart B has mass has mass mB= 2 mAand is initially at rest. Immediately after the collision, the incident cart has final speed vA,2and the target cart has final speed vB,2.Calculate the final velocities of the carts as a function of the initial speed .Example: Elastic Collision in 2-dIn the laboratory reference frame, an “incident” particle with mass m1, is moving with given initial speed v10. The second “target” particle is of mass m2and at rest. After an elastic collision, the first particle moves off at an angle θ1,fwith respect to the initial direction of motion of the incident particle with final speed v1,f. Particle two moves off at an angle θ2,fwith final speed v2,f. (i) Find the equations that represent conservation of momentum and energy. Assume no external forces. (ii) Design a strategy to find v1,f, v2,f, and θ2,f. (You will solve this type of problem on your homework.)4Momentum AnalysisSince momentum is a vector quantity, identify the initial and final vector components of the total momentumInitial State:x-comp: y-comp:Final State:x-comp: y-comp:sys011,0m=pvGGsys11, 2 2,fffmm=+pvvGGG px,0sys= m1v1,0 py,0sys= 0 px, fsys= m1v1, fcosθ1, f+ m2v2, fcosθ2, f py, fsys= m1v1, fsinθ1, f− m2v2, fsinθ2, fConservation Laws AnalysisNo external forces are acting on the system:Collision is elastic:py,0sys= py, fsyspx,0sys= px, fsysm1v1,0= m1v1, fcosθ1, f+ m2v2, fcosθ2, f0 = m1v1, fsinθ1, f− m2v2, fsinθ2, f12m1v1,02=12m1v1, f2+12m2v2, f2Strategy:• Three unknowns: v1f, v2f, and θ2,f• First squaring then adding the momentum equations and equations and solve for v2fin terms of v1f.• Substitute expression for v2fkinetic energy equation and solve quadratic equation for v1f• Use result for v1fto solve expression for v2f• Divide momentum equations to obtain expression for θ2,fExperiment 4: Momentum and

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