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Elastic Collisions Momentum and Kinetic Energy An object in motion has a momentum based on its mass and velocity. • p = mv The object also has kinetic energy. • K = ½ mv2 = p2 / 2m Kinetic Energy at Collision m1 Energy is conserved only for conservative forces. v1i v2i m2 Before: K i 12 m1v12i 12 m2 v22i Internal forces may be nonconservative. The force at the collision is not always conservative. energy lost to heat v1f After: v2f K f 12 m1v12f 12 m2v22 f Elastic Collision For conservative forces the energy is conserved. Elastic After the collision of contact the potential energy is zero. The total kinetic energy is conserved – equal before and after the collision. This an elastic collision. Pi Pf Ki K f Double Conservation Elastic collisions conserve both momentum and kinetic energy. Two equations govern all elastic collisions. m1 v1i m1 m2 v2i before m1v1i m2 v2i m1v1 f m2 v2 f 1 2 m1v12i 12 m2 v22i 12 m1v12f 12 m2 v22 f m2 v1f v2f after Head-on Collision An elastic head-on collision takes place in one dimension. v1i v2i If the collision is not headon, the force pair is in a different direction. v1i v2i m1 m2 force and velocity in a line m1 m2 force and velocity on different lines Related Velocities momentum in a line solve for velocities m1v1i m1v1 f m2v2 f m2v2i v1i v1 f v2 f v2i m1 (v1i v1 f ) m2 (v2 f v2i ) kinetic energy conservation 1 2 m1v12i 12 m1v12f 12 m1v22 f 12 m2 v22i m1 (v12i v12f ) m2 (v22 f v22i ) m1 (v1i v1 f )(v1i v1 f ) m2 (v2 f v2i )(v2 f v2i ) v1i v2i v2 f v1 f v1i v2i m1 m2 Equal Masses A 150 g ball moves at 1.4 m/s. • The momentum is 0.21 kg m/s v1i m1 It strikes an equal mass ball at rest. • • • • v1i = 1.4 m/s v2i = 0 Therefore, v1f = 0 and v2f = v1i m2 m1 m2 v2f momentum: v1i v2i v2 f v1 f kinetic energy: v1i v2i v2 f v1 f Striking a Heavy Mass Let m1 << m2, when a golf ball bounces off the floor. The floor is at rest. • v2i = 0 v1i m1v1i m2 v2 f m1v1 f kinetic energy: v1i v2 f v1 f The final velocity is equal and opposite the initial velocity m1 momentum: v1f combined: v1 f m1 m2 v1i v1i m1 m2 v2 f 0 Striking a Light Mass Let m1 >> m2, when a car strikes a ball. The ball is at rest. v1i v2 f v1 f For a very heavy m1 , the final velocity of m2 is twice the initial velocity of m1 . v2f m2 v1i m1v1i m2 v2 f m1v1 f kinetic energy: • v2i = 0 momentum: combined: m1 m2 v1 f v1i v1i m1 m2 v2 f 2v1i