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• Momentum Impulse Conservation of Momentum Collisions Click one Momentum • Momentum is the product of mass times velocity of an object P = m v • Momentum is a vector quantity (same direction as v) Units = kg m/s NEWTON’S LAWS OF MOTION Second LAW: The force on an object is equal to the product of that object’s mass times its acceleration. The acceleration is in the same direction as the force. F=m.a a = Dv/Dt F = m . Dv/Dt F . Dt = m . Dv Impulse =F . Dt Dp = m . Dv Change in “momentum” A baseball of mass 0.14 kg is moving at 35 m/s. a. find the momentum of the baseball b. find the velocity at which a bowling ball, mass 7.26 kg, would have the same momentum as the baseball Given: mass = 0.14 kg velocity = 35 m/s unknown = p NEWTON’S LAWS OF MOTION Third LAW: For every force that one object exerts on a second object, there is an equal but oppositely directed force that the second object exerts on the first object. (For every action there is an equal but opposite reaction) F . Dt = m . Dv If the external force acting on a system of objects is zero, the total momentum is conserved. CONSERVATION OF MOMENTUM During a collision, there are no external forces, so momentum is conserved. This means: total momentum before collision = total momentum after collision The Conservation of Momentum The law of conservation of momentum states: The momentum of any closed, isolated system does not change. This means that the momentum before a collision is equal to the momentum after a collision. p before Collision = p after Example Glider “A” of mass 0.355 kg moves along a frictionless air track with a velocity of 0.095 m/s. It collides with glider “B” of mass 0.710 kg moving in the same direction at a speed of 0.045 m/s. After the collision, glider “A” continues in the same direction with a velocity of 0.035 m/s. What is the velocity of glider “B” after the collision? Types of Collisions • Elastic - no kinetic energy is lost during collisions (things bounce off each other) • Partially Inelastic - some kinetic energy is lost during collisions • Perfectly Inelastic - objects stick together Example Collision: Total momentum before collision = Total momentum after collision v = 10 m/s v = 0 m/s Before: V’ = 5 m/s After: (m 10 m/s)before = (2m V’)after This is an example of an inelastic collision Test your understanding: A 1-kg cart and a 2-kg cart roll toward the center of a straight track from opposite ends of the track, each with a speed of 1 m/s. They collide and stick. The combined mass moves at a speed of (A) 0 m/s. (B) 1/2 m/s. (C) 1/3 m/s. (D) 1/6 m/s. (E) 1.5 m/s. 1 m/s 1 m/s Before: 1 m/s 1 m/s V=? After: (2 kg 1 m/s - 1 kg 1 m/s)before = (3 kg V)after V = 1/3 m/s Before the collision, the ball has momentum and the person does not. The collision causes the ball to lose momentum and the person to gain momentum. After the collision, the ball and the person travel with the same velocity ("v") across the ice. Before Collision After Collision Person Medicine ball Total 0 300 300 60 * v 15 * v 300 60*v + 15*v = 300 75*v = 300 v = 4 km/hr Truck Before Collision 3000 * 10 = 30 000 After Collision 3000 * v Car Total 0 30 000 1000 * 15 = 15 000 30 000 3000*v + 15 000 = 30 000 3000*v = 15 000 v = 5.0 m/s Conservation of Momentum in Two Dimensions A 2.0 kg ball, A, is moving at a velocity of 5.0 m/s. It collides with a stationary ball, B, also of mass 2.0 kg. After the collision, ball A moves off in a direction 300 to the left of its original direction. Ball B moves off in a direction 900 to the right of ball A’s final direction. a. Draw a vector diagram to find the momentum of ball A and of ball B after the collision b. Find the velocities of the balls after the collision a. pA = 8.66 kg m/s pB = 5 kg m/s b. vA’ = 4.33 m/s vB’ = 2.5 m/s a. pA = 8.66 kg m/s pB = 5 kg m/s b. vA’ = 4.33 m/s vB’ = 2.5 m/s Example of Elastic Collisions Behavior: Number of balls in always equals the number of balls out. What accounts for the behavior of this system of swinging balls? How high up will the struck ball go? h Behavior of balls is consistent with conservation of momentum Total momentum before collision: mv = 0+0+0+0+mv Total momentum after collision: mv = mv+0+0+0+0 Other possibilities also conserve momentum 2 balls in, one ball out Momemtum before collision: 2mv = 0+0+0+(2m)v Momentum after collision: 2mv = m(2v)+0+0+0+0 momentum is conserved Energy Conservation Elastic collision KE is conserved Case: two balls in, one ball out with twice the speed Energy is not conserved! There is only one case where BOTH energy and momentum are conserved