2.3 Momentum Momentum: 𝑝=𝑚𝑣 (unit: kg m s-1) Alternative definition of N2: 𝐹=𝑚𝑎= ∆𝑝 𝑡 Impulse: change in momentum, ∆𝑝=𝐹𝑡 Area under force-time-graph gives the impulse Momentum is ALWAYS conserved!!!

Ex 1: What is the momentum of a hockey puck with mass 170 g moving at 10 m s-1?

Ex 2: Which action requires more force, to block the puck (m = 0,17 kg, u = 10 m s-1) or to catch it? Both actions take 0.10 s. https://im.mtv.fi/image/4508284/landscape16_9/1024/576/6b96fc3dd44ef2b34d3cb0cbd63bd9c2/ea/atte-engren-venaja-ottelussa.jpg (041217)

Ex. 3: A force of 100 N from a racket acts on a tennis ball (m = 0 Ex. 3: A force of 100 N from a racket acts on a tennis ball (m = 0.090 kg, u =20 ms-1) for 0.045 s. Calculate: The impulse The change in velocity The final velocity

Collisions Total momentum before and after the collision are the same! Inelastic collisions: (normal case) Momentum is conserved Formula: 𝑚 1 𝑢 1 + 𝑚 2 𝑢 2 = 𝑚 1 𝑣 1 + 𝑚 2 𝑣 2 Elastic collisions: (special case 1) ”Perfect bounce” Momentum and kinetic energy conserved Totally inelastic collisions: (special case 2) ”Stick together” Formula becomes: 𝑚 1 𝑢 1 + 𝑚 2 𝑢 2 = (𝑚 1 + 𝑚 2 )𝑣 The direction is important!

Example 1: An ice skater (m = 49 kg) at rest catches a ball (m = 1 kg) with velocity 10 m s-1. What happens? Totally inelastic: 𝑚 1 𝑢 1 + 𝑚 2 𝑢 2 = (𝑚 1 + 𝑚 2 )𝑣 𝑚 1 =49 kg, 𝑢 1 =0 m s-1 𝑚 2 =1 kg, 𝑢 2 =10 m s-1 49 kg∙0 m s −1 +1 kg∙10 m s −1 =(49 kg+1 kg)∙𝑣 𝑣= 1 kg∙10 m s −1 49 kg+1 kg =0.2 m s −1 Answer: the skater (holding the ball) will start moving with 0.2 m s-1 in the same direction as the ball that hit her

Example 2: A car (m = 1000 kg, v = 100 km h-1) and a truck (m = 15 000 kg, v = 80 km h-1) collide head on and stick together. What happens? Totally inelastic: 𝑚 1 𝑢 1 + 𝑚 2 𝑢 2 = (𝑚 1 + 𝑚 2 )𝑣 𝑚 1 =1000 kg, 𝑢 1 =100 km h-1 𝑚 2 =15 000 kg, 𝑢 2 = - 80 km h-1 1000 kg∙100 km h −1 −15000 kg∙80 km h −1 =(1000 kg+15000 kg)∙𝑣 𝑣= 1000 kg∙100 km h −1 −15000 kg∙80 km h −1 1000 kg+15000 kg =−68.75 km h −1 Answer: The combined wreck will move at 69 km h-1 in the direction of the truck

Example 3: A car (m = 1000 kg, v = 100 km h-1) and a motor cycle (m = 250 kg, v = 80 km h-1) collide head on. Right after the collision the velocity of the car is 60 km h-1. What happens with the motor cycle? Inelastic: 𝑚 1 𝑢 1 + 𝑚 2 𝑢 2 = 𝑚 1 𝑣 1 + 𝑚 2 𝑣 2 𝑚 1 =1000 kg, 𝑢 1 =100 km h-1, 𝑣 1 =60 km h-1 𝑚 2 =250 kg, 𝑢 2 = - 80 km h-1 1000 kg∙100 km h −1 −250 kg∙80 km h −1 =1000 kg∙60 km h −1 +250 kg∙ v 2 𝑣= 1000 kg∙100 km h −1 −250 kg∙80 km h −1 −1000 kg∙60 km h −1 250 kg =80 km h −1 Answer: The motor cycle bounces back at 80 km h-1 in the direction of the car

Example 4: A Volvo (m = 2000 kg, v = 50 km h-1 from the South) and a Nissan (m = 1500 kg, v = 40 km h-1 from the West) collide at a crossroads and stick together. What happens? Totally inelastic in 2D: 𝑚 1 𝑢 1𝑥 + 𝑚 2 𝑢 2𝑥 = (𝑚 1 + 𝑚 2 ) 𝑣 𝑥 𝑚 1 𝑢 1𝑦 + 𝑚 2 𝑢 2𝑦 = (𝑚 1 + 𝑚 2 ) 𝑣 𝑦 𝑚 1 =2000 kg, 𝑢 1𝑥 =0 km h-1, 𝑢 1𝑦 =50 km h-1 𝑚 2 =1500 kg, 𝑢 2𝑥 = 40 km h-1, 𝑢 1𝑦 =0 km h-1 X-axis: 𝑣 𝑥 = 1500 𝑘𝑔∙40 𝑘𝑚 ℎ −1 2000 𝑘𝑔+1500 𝑘𝑔 =17.14…𝑘𝑚 ℎ −1 Y-axis: 𝑣 𝑥 = 2000 𝑘𝑔∙50 𝑘𝑚 ℎ −1 2000 𝑘𝑔+1500 𝑘𝑔 =28.57…𝑘𝑚 ℎ −1 Total velocity: 𝑣= 17.14…𝑘𝑚 ℎ −1 2 + 28.57…𝑘𝑚 ℎ −1 2 =33.31…𝑘𝑚 ℎ −1 Angle: 𝛼= tan −1 28.57…𝑘𝑚 ℎ −1 17.14…𝑘𝑚 ℎ −1 =59.0…° Answer: The combined wreck continues at 33 km h-1 at an angle of 59° North of East