Mrs. Todd’s Honors Physics Three ways to solve elastic collision problems Elastic collision problems can be a pain to solve. Since there are often two unknowns, v1f and v2f, then you need to have two equations to solve simultaneously. 1. Traditional Method – use the following two equations and solve simultaneously m1v1i + m2 v2i = m1v1 f + m2 v2 f ← Conservation of Momentum 1 2 1 2 1 2 1 2 2 m1v1i + 2 m2 v2i = 2 m1v1 f + 2 m2 v2 f ← Conservation of Kinetic Energy This method could be time consuming because of the squared terms in the conservation of kinetic energy formula. However, this is probably the easiest method to conceptualize and there are no new formulas to memorize. 2. Approach and Recess Velocities Method – use the following two equations and solve simultaneously m1v1i + m2 v2i = m1v1 f + m2 v2 f ← Conservation of Momentum v1i − v2i = v2 f − v1 f ← Approach / Recess Velocities The second equation here was derived using both the conservation of momentum and the conservation of energy formulas, but it can be easier to use than the conservation of kinetic energy formula because of the lack of squared terms. 3. Center of Mass Reference Frame Method – use the following procedure 1. Identify the initial velocities: v1i and v2i m1v1i + m2v2i 2. Find the velocity of the center of mass using vcm = m1 + m2 3. Transform the velocities to center of mass reference frame: u1i = v1i − vcm and u2i = v2i − vcm 4. Find the final velocities: u1 f = −u1i and u2 f = −u2i 5. Transform these velocities back to lab reference frame: v1 f = u1 f + vcm and v2 f = u2 f + vcm This method utilizes the fact that the velocity of the center of mass remains constant before and after the collision since there is no net external force on the system. This is probably the easiest method computationally. In fact, it can be further simplified by using the following very short procedure: 1. Identify the velocities: v1i and v2i m1v1i + m2v2i 2. Find the velocity of the center of mass using vcm = m1 + m2 3. Find the final velocities: v1 f = −v1i + 2vcm and v2 f = −v2i + 2vcm .