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SPH4U: Lecture 15 Today’s Agenda l Elastic collisions in one dimension Analytic solution (this is algebra, not rocket science!) l Center of mass reference frame è Colliding carts problem è Two dimensional collision problems (scattering) è Solving elastic collision problems using COM and inertial reference frame transformations l Some interesting properties of elastic collisions è Center of mass energy and energy of relative motion

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Momentum Conservation: Review l The concept of momentum conservation is one of the most fundamental principles in physics. l This is a component (vector) equation. è We can apply it to any direction in which there is no external force applied. l You will see that we often have momentum conservation even when kinetic energy is not conserved.

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Comment on Energy Conservation We have seen that the total kinetic energy of a system undergoing an inelastic collision is not conserved. è Mechanical Energy is lost: (remember what this is??) »Heat (bomb) »Bending of metal (crashing cars) Kinetic energy is not conserved since dissipative work is done during an inelastic collision! (here, KE equals mechanical energy) l Total momentum, P T, along a certain direction is conserved when there are no external forces acting in this direction. è F = ma = dP T /dt says this has to be true!! (Newton’s Laws) è In general, momentum conservation is easier to satisfy than mechanical energy conservation. è Remember: in the absence of external forces, total energy (including heat…) of a system is always conserved even when mechanical energy is not conserved. è How much do two objects that inelastically collide heat up?

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Lecture 15, Act 1 Collisions l A box sliding on a frictionless surface collides and sticks to a second identical box which is initially at rest. l What is the ratio of initial to final kinetic energy of the system? (a) (a) 1(b) (c) (c) 2

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Lecture 15, Act 1 Solution l No external forces in the x direction, so P X is constant. v mm mmv / 2 x

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Lecture 15, Act 1 Solution l Compute kinetic energies: v mm mmv / 2

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Lecture 15, Act 1 Another solution l We can write mm mm l P is the same before and after the collision. l The mass of the moving object has doubled, hence the kinetic energy must be half.

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Lecture 15, Act 1 Another Question: l Is it possible for two blocks to collide inelastically in such a way that the kinetic energy after the collision is zero?

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Lecture 15, Act 1 Another Question l Is it possible for two blocks to collide inelastically in such a way that the kinetic energy after the collision is zero? YES: If the CM is not moving! CM

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Elastic Collisions l Elastic means that kinetic energy is conserved as well as momentum. l This gives us more constraints è We can solve more complicated problems!! è Billiards (2-D collision) è The colliding objects have separate motions after the collision as well as before. è all 3D collision problems can be solved in 2 dimensions by using center of mass inertial reference frame l Start with a simpler 1-D problem Initial Final

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Elastic Collision in 1-D what has to happen v 1,i v 2,i initial x m1m1 m2m2 v 1,f v 2,f final m1m1 m2m2 Why is this elastic? Kinetic energy potential energy kinetic energy The spring is conservative Maybe, it depends…

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Elastic Collision in 1-D the spring is conservative x Conserve P X : (no external forces!) m 1 v 1,i + m 2 v 2,i = m 1 v 1,f + m 2 v 2,f Conserve Kinetic Energy: (it’s elastic!) 1 / 2 m 1 v 2 1,i + 1 / 2 m 2 v 2 2,i = 1 / 2 m 1 v 2 1,f + 1 / 2 m 2 v 2 2,f Suppose we know v 1,i and v 2,i We need to solve for v 1,f and v 2,f Should be no problem 2 equations & 2 unknowns! v 1,i v 2,i before m1m1 m2m2 v 1,f v 2,f after

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Elastic Collision in 1-D l However, solving this can sometimes get a little bit tedious since it involves a quadratic equation!! l A simpler approach is to introduce the Center of Mass Reference Frame l First, describe the solution to the problem using algebra. è Useful analysis and useful formulae m 1 v 1,i + m 2 v 2,i = m 1 v 1,f + m 2 v 2,f 1 / 2 m 1 v 2 1,i + 1 / 2 m 2 v 2 2,i = 1 / 2 m 1 v 2 1,f + 1 / 2 m 2 v 2 2,f momentum energy

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Elastic Collision in 1-D special case: equal masses l If the masses of the two objects are equal the algebra is not too bad. Let’s see what we get… l Divide through by m = m 1 = m 2 m 1 v 1,i + m 2 v 2,i = m 1 v 1,f + m 2 v 2,f 1 / 2 m 1 v 2 1,i + 1 / 2 m 2 v 2 2,i = 1 / 2 m 1 v 2 1,f + 1 / 2 m 2 v 2 2,f momentum energy v 1,i + v 2,i = v 1,f + v 2,f v 2 1,i + v 2 2,i = v 2 1,f + v 2 2,f

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Elastic Collision in 1-D special case: equal masses l Now just rearrange equations to bring v 1,i and v 1,f to left hand side, and v 2,i and v 2,f to rhs l Divide through energy equation by momentum equation which gives Particles just trade velocities in 1-D elastic collision of equal mass objects (let equation talk to you…) momentum energy v 1,i + v 1,f = v 2,f + v 2,i v 1,i - v 1,f = v 2,f - v 2,i v 2 1,i - v 2 1,f = v 2 2,f - v 2 2,i (v 1,i - v 1,f )(v 1,i + v 1,f ) = (v 2,f - v 2,i )(v 2,f + v 2,i ) v 2,f = v 1,i v 1,f = v 2,i v 1,i - v 1,f = v 2,f - v 2,i

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Elastic Collision in 1-D general case: unequal masses l Conserve linear momentum and mechanical energy, but now the masses are different: l Divide through energy equation by momentum equation which gives Now solving these two linear equations is only a bit more complicated momentum energy v 1,i + v 1,f = v 2,f + v 2,i m 1 (v 1,i - v 1,f ) = m 2 (v 2,f - v 2,I ) m 1 (v 2 1,i - v 2 1,f ) = m 2 (v 2 2,f - v 2 2,I ) m 1 (v 1,i - v 1,f )(v 1,i + v 1,f ) =m 2 (v 2,f - v 2,i )(v 2,f + v 2,i ) m 1 (v 1,i - v 1,f ) = m 2 (v 2,f - v 2,I )

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Elastic Collision in 1-D general case: unequal masses l Algebra just gave us the following equations based on conservation of momentum and mechanical energy: l Now just solve for final velocities, v 1,f and v 2,f in terms of v 1,i and v 2,i v 1,i + v 1,f = v 2,f + v 2,i m 1 (v 1,i - v 1,f ) = m 2 (v 2,f - v 2,I ) v 2,f = v 1,i v 1,f = v 2,i When m 1 = m 2

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Another way to solve elastic collision problems: CM Reference Frame l We have shown that the total momentum of a system of particles is the velocity of the CM times the total mass: PV P NET = MV CM. l We have also discussed reference frames that are related by a constant velocity vector (i.e.they’re in relative motion). l Now consider putting yourself in a reference frame in which the CM is at rest. We call this the CM reference frame. V P è In the CM reference frame, V CM = 0 (by definition) and therefore P NET = 0. è This is a cool mathematical tool that makes the algebra solving this much simpler è (it doesn’t change the physical situation)

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Lecture 15, Act 2 Force and Momentum l Two men, one heavier than the other, are standing at the center of two identical heavy planks. The planks are at rest on a frozen (frictionless) lake somewhere in Ontario. l The men start running on their planks at the same speed. è Which man is moving faster with respect to the ice? (a) (b) (c) (a) heavy (b) light (c) same

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Lecture 15, Act 2 Conceptual Solution l The external force in the x direction is zero (frictionless): è The CM of the systems can’t move! è Aha! this is the key!! x X X X X CM

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Lecture 15, Act 2 Conceptual Solution l The external force in the x direction is zero (frictionless): è The CM of the systems can’t move! l The men will reach the end of their planks at the same time, but lighter man will be further from the CM at this time. è His motion doesn’t count as much, since he is less massive l The lighter man moves faster with respect to the ice! X X X X CM

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Lecture 15, Act 2 Algebraic Solution l Consider one of the runner-plank systems: l There is no external force acting in the x-direction: è Momentum is conserved in the x-direction! è The initial total momentum is zero, hence it must remain so. è We are observing the runner in the CM reference frame! x l Let the mass of the runner be m and the plank be M. m M l Let the speed of the runner and the plank with respect to the ice be v R and v P respectively. vRvR vPvP

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Lecture 15, Act 2 Algebraic Solution l The velocity of the runner with respect to the plank is V = v R - v P (same for both runners). x m M vRvR vPvP Mv P = - mv R (momentum conservation, it’s zero!) Plugging v P = v R - V into this we find: So v R is greater if m is smaller.

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Example 1: Using CM Reference Frame l A glider of mass m 1 = 0.2 kg slides on a frictionless track with initial velocity v 1,i = 1.5 m/s. It hits a stationary glider of mass m 2 = 0.8 kg. A spring attached to the first glider compresses and relaxes during the collision, but there is no friction (i.e. energy is conserved). What are the final velocities? x m1m1 m2m2 v 1,i v v 2,i = 0 V V CM m1m1 m2m2 m1m1 v v 1,f v v 2,f m2m2 + = CM

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Example 1... l Four step procedure V è First figure out the velocity of the CM, V CM. »Vvvv »V CM = (m 1 v 1,i + m 2 v 2,i ), but v 2,i = 0 in this case so Vv V CM = v 1,i V l So V CM = 1 / 5 (1.5 m/s) = 0.3 m/s v (for v 2,i = 0 only)

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Example 1... V v v* l If the velocity of the CM in the “lab” reference frame is V CM, and the velocity of some particle in the “lab” reference frame is v, then the velocity of the particle in the CM reference frame is v* where: v*vVv*,vV v* = v - V CM (where v*, v, V CM are vectors) V V CM v v* v* This is the “lab” frame velocity This is the CM frame velocity If you were traveling along with the CM, you would see the velocity of the mass to be less than in the lab frame in this case

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Example 1... è Calculate the initial velocities in the CM reference frame (all velocities are in the x direction): v*vV v* 1,i = v 1,i - V CM = 1.5 m/s m/s = 1.2 m/s v*vV v* 2,i = v 2,i - V CM = 0 m/s m/s = -0.3 m/s v* v* 1,i = 1.2 m/s v* v* 2,i = -0.3 m/s

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Example 1 continued... V l Now consider the collision viewed from a frame moving with the CM velocity V CM. ( jargon: “in the CM frame”) m1m1 m2m2 v* v* 1,i v* v* 2,i x m2m2 m1m1 m1m1 v* v* 1,f v* v* 2,f m2m2 Movie

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Energy in Elastic Collisions: l Use energy conservation to relate initial and final velocities. The total kinetic energy in the CM frame before and after the collision is the same, it’s elastic!! (look how we write this…) l But the total momentum is zero, both initial and final: l So: (and the same for particle 2) Therefore, in 1-D: v* 1,f = -v* 1,i v* 2,f = -v* 2,i Likewise for final v’s

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Example 1... v* v* 1,i v* v* 2,i x m1m1 m2m2 m1m1 m2m2 v*v* v* 1,f = - v* 1,i = -1.2m/s v*v* v* 2,f = - v* 2,i =.3 m/s m1m1 m2m2 v*v*v*v* v* 1,f = -v* 1,i v* 2,f = -v* 2,i Calculate the final velocities in the CM frame:

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Example 1... è Now we can calculate the final velocities in the lab reference frame, using: vv*V v 1,f = v* 1,f + V CM = -1.2 m/s m/s = -0.9 m/s vv*V v 2,f = v* 2,f + V CM = 0.3 m/s m/s = 0.6 m/s v v 1,f = -0.9 m/s v v 2,f = 0.6 m/s v v*V v = v* + V CM v*vV v* = v - V CM Four easy steps! No need to solve a quadratic equation!! Especially important in 2D v*v* v* f = -v*,i

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Lecture 15, Act 3 Moving Between Reference Frames l Two identical cars approach each other on a straight road. The red car has a velocity of 40 mi/hr to the left and the green car has a velocity of 80 mi/hr to the right. è What are the velocities of the cars in the CM reference frame? (a) V RED = - 20 mi/hr (b) V RED = - 20 mi/hr (c) V RED = - 60 mi/hr V GREEN = + 20 mi/hr V GREEN = +100 mi/hr V GREEN = + 60 mi/hr

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Lecture 15, Act 3 Moving Between Reference Frames l The velocity of the CM is: x = 20 mi / hr 20mi/hr CM 80mi/hr 40mi/hr l So V GREEN,CM = 80 mi/hr - 20 mi/hr = 60 mi/hr l So V RED,CM = - 40 mi/hr - 20 mi/hr = - 60 mi/hr l The CM velocities are equal and opposite since P NET = 0 !!

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l As a safety innovation, Volvo designs a car with a spring attached to the front so that a head on collision will be elastic. If the two cars have this safety innovation, what will their final velocities in the lab reference frame be after they collide? Lecture 15, Act 3 Aside 20mi/hr CM 80mi/hr 40mi/hr x

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Lecture 15, Act 3 Aside Solution v*v*v*v* v* GREEN,f = -v* GREEN,i v* RED,f = -v* RED,i v*v* v* GREEN,f = -60 mi/hr v* RED,f = 60 mi/hr v´ v*V v´ = v* + V CM v´ v´ GREEN,f = -60 mi/hr + 20 mi/hr = - 40 mi/hr v´ v´ RED,f = 60 mi/hr + 20 mi/hr = 80 mi/hr v* v* GREEN,i = 60 mi/hr v* v* RED,i = -60 mi/hr

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Summary: Using CM Reference Frame : Determine velocity of CM : Calculate initial velocities in CM reference frame : Determine final velocities in CM reference frame : Calculate final velocities in lab reference frame V V CM = v*vV v* = v - V CM v* f = -v* i v v*V v = v* + V CM vv (m 1 v 1,i + m 2 v 2,i )

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Interesting Fact l We just showed that in the CM reference frame the speed of an object is the same before and after the collision, although the direction changes. l The relative speed of the blocks is therefore equal and opposite before and after the collision. ( v* 1,i - v* 2,i ) = - ( v* 1,f - v* 2,f ) l But since the measurement of a difference of speeds does not depend on the reference frame, we can say that the relative speed of the blocks is therefore equal and opposite before and after the collision, in any reference frame. è Rate of approach = rate of recession v* 1,i v* 2,i v* 1,f = -v* 1,i v* 2,f = -v* 2,i This is really cool and useful too!

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Recap of lecture Elastic Collision –a collision in which the total kinetic energy after the collision equals the total kinetic energy before the collision. If m 2 is initially at rest, then you can use: conservation of energy and conservation of momentum, or:

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Example A ball with mass 2.5x10 -2 kg moving at 2.3 m/s collides with a stationary 2.0x10 -2 kg ball. If the collision is elastic, determine the velocity of each ball after the collision.

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Solution K.E. Momentum Given Re-arranging the Momentum equation with v 2i =0, we obtain:

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Substitute this into the K.E. equation: Substitute in: Note: Both balls end up moving in the same direction.

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Solution by CM A ball with mass 2.5x10 -2 kg moving at 2.3 m/s collides with a stationary 2.0x10 -2 kg ball. If the collision is elastic, determine the velocity of each ball after the collision. Given

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