Lecture 14: Collisions & Momentum

Questions of Yesterday A 50-kg object is traveling with a speed of 100 m/s and a 100-kg object is traveling at a speed of 50 m/s. 1a) Which object has more momentum? 1b) Which object has more kinetic energy? a) 50-kg object b) 100-kg object c) they are equal 2) Would a head-on collision between two cars be more damaging to the occupants if the cars stuck together or if the cars rebounded upon impact? a) if the cars stuck together b) if the cars rebounded c) both collisions would be equally damaging d) it depends on the relative masses of the cars

Collisions MOMENTUM of an object is CONSERVED if F net = 0 What happens in a collision? mv i1 + mv i2 = mv f1 + mv f2 If no net external force acts on a system of objects… The total momentum of the system remains constant in time p i = p f

Kinetic Energy Conservation Is the total kinetic energy conserved in a collision? Is it possible to lose kinetic energy? How? Kinetic energy is generally NOT conserved in a collision Kinetic Energy Sound, Heat, Deformation, Kinetic Energy

Types of Collisions INELASTIC Collisions Momentum is Conserved Kinetic Energy is NOT Conserved p i = p f KE i > KE f PERFECTY INELASTIC Collisions Momentum is Conserved Kinetic Energy is NOT Conserved Objects stick together v f1 = v f2 ELASTIC Collisions Momentum is Conserved Kinetic Energy IS Conserved p i = p f KE i = KE f

Perfectly Inelastic Collisions Momentum is Conserved Kinetic Energy is NOT Conserved Objects stick together v f1 = v f2 p i1 = m 1 v i1 p i2 = m 2 v i2

Perfectly Inelastic Collisions Momentum is Conserved Kinetic Energy is NOT Conserved Objects stick together v f1 = v f2 p i1 = m 1 v i1 p i2 = m 2 v i2 vfvf p i = p f

Perfectly Inelastic Collisions p i = p f m 1 v i1 + m 2 v i2 = ( m 1 + m 2 ) v f p i1 = m 1 v i1 p i2 = m 2 v i2 vfvf v f = m 1 v i1 + m 2 v i2 ( m 1 + m 2 )

Perfectly Inelastic Collisions p i = p f m 1 v i1 + m 2 v i2 = ( m 1 + m 2 ) v f v f = m 1 v i1 + m 2 v i2 ( m 1 + m 2 ) Vector Equations! Velocities must be in same direction!! Equations for 1D collisions! In general: p ix = p fx AND p iy = p fy

Perfectly Inelastic Collisions p i = p f m 1 v i1 + m 2 v i2 = ( m 1 + m 2 ) v f How much Kinetic Energy is lost in a perfectly inelastic collision?  KE = KE f - KE i KE f = (1/2)(m 1 + m 2 )v f 2 KE i = KE i1 + KE i2 = (1/2)m 1 v i1 2 + (1/2)m 2 v i2 2

Perfectly Inelastic Collisions A 10.0 g bullet is fired horizontally into a 100-g wooden block that is initially at rest on a frictionless horizontal surface and connected to a spring having spring constant 100 N/m. The bullet becomes embedded in the block. If the bullet-block system compresses the spring by a maximum of 100 cm.. What was the initial speed of the bullet at impact with the block?

Elastic Collisions 2 equations -> Can solve for 2 unknown quantities How do the final velocities of the objects compare with the initial velocities? Momentum is Conserved: p i = p f Kinetic Energy IS Conserved: KE i = KE f (1/2)m 1 v i1 2 + (1/2)m 2 v i2 2 = (1/2)m 1 v f1 2 + (1/2)m 2 v f2 2 v i1 - v i2 = -(v f1 - v f2 ) m 1 v i1 + m 2 v i2 = m 1 v f1 + m 2 v f2

Elastic Collisions A 10-kg object moving to the right at 20.0 m/s makes an elastic head-on collision with a 20.0-kg object moving in the opposite direction at 30.0 m/s. What is the velocity of each object after the collision? What is the change in the kinetic energy of each object? What is the change in kinetic energy of the system?

2D (Glancing) Collisions x-component: m 1 v i1x + m 2 v i2x = m 1 v f1x + m 2 v f2x y-component: m 1 v i1y + m 2 v i2y = m 1 v f1y + m 2 v f2y Momentum is a VECTOR In 2 Dimensional collisions… momentum in EACH direction is conserved! (1/2)m 1 v i1 2 + (1/2)m 2 v i2 2 = (1/2)m 1 v f1 2 + (1/2)m 2 v f2 2 Kinetic Energy is a SCALAR quantity Only the speeds of the objects are important

2 Dimensional Collisions A 0.5 kg puck, initially at rest on a frictionless horizontal surface, is struck by a 0.25-kg puck that is initially moving along the x-axis with a velocity of 2.0 m/s. After the collision the 0.25-kg puck has a speed of 1.0 m/s at an angle of 30 o to the positive x-axis. What is the velocity of the 0.5-kg puck after the collision? Is this collision elastic? If not, what is the fraction of kinetic energy lost in the collision?

Questions of the Day 1) A piece of clay traveling north with speed v collides perfectly inelastically with an identical piece of clay traveling east with speed v. What direction does the resultant piece of clay travel? a) north b) east c) 45 o N of E d) 45 o S of W 2) If Ball 1, moving with an initial speed v, collides with Ball 2 which is initially at rest, which scenario is not possible following the collision? a) Both balls are moving b) Ball 1 is at rest and Ball 2 is moving c) Ball 2 is at rest and Ball 1 is moving d) Both balls are at rest