Momentum and Collisions

Unit 6

Momentum- (inertia in motion)• Momentum describes an object’s

motion• Momentum equals an object’s mass

times its velocity• p = mv• Units: kg.m/s

• If both a truck and a roller skate are rolling down a hill with the same speed which has more momentum?

• If the truck is at rest and the roller skate is moving, which has more momentum?

Both an 18-wheeler and a compact car are traveling down the highway at 60 mph, which has the greatest momentum? a. 18-wheelerb. compact carc. Bothd. Neither

• Impulse (also “change in momentum”) occurs when a force is applied for a time interval

• Impulse equals Force times time

• J = F∙ t

• Units are the same as momentum, kg ∙m/s or may also be N∙s

Force and Impulse

J = F•t = p = mvf – mvi

Examples: •golf club hitting a golf ball•Baseball bat hitting a baseball• car hitting a haystack•Car hitting a brick wall

• To increase the momentum of an object, apply the greatest force possible for as long as possible.

• A golfer teeing off and a baseball player trying for a home run do both of these things when they swing as hard as possible and follow through with their swing.

Decreasing Momentum

• If you were in a car that was out of control and had to choose between hitting a haystack or a concrete wall, you would choose the haystack.

• If the change in momentum occurs over a long time, the force of impact is small

• If the change in momentum occurs over a short time, the force of impact is large.

•When hitting either the wall or the haystack and coming to a stop, the change in momentum (or impulse) is the same.

•The same impulse does not mean the same amount of force or the same amount of time.•It means the same product of force and time.•To keep the force small, we extend the time.

F, t, and p relationship• When you extend the time, you reduce the force.• Used for safety equipment and sports equipment

• A padded dashboard in a car is safer than a rigid metal one.

• Airbags save lives.• To catch a fast-moving ball, extend your hand

forward and move it backward after making contact with the ball.

p is the same; there is an inverse relationship between force and time

Consider this!

• Compare the magnitude of force required to stop a 70 kg passenger moving at 25 m/s in 0.75 s (airbag) and 0.026 s (dashboard)

• What is the advantage of airbags in our cars?

BouncingImpulse is greater when bouncing occurs.The force needed to cause an object to bounce is greater than the force needed when it does not bounce.Ex. A glass bottle falls on your head. If it breaks when it hits your head, the impulse has ended. If it bounces off your head, your head applies a force to send it back up.

Stopping Times and Distances• Depends on the impulse-momentum theorem and the

Work-KE theorem • Be careful when distinguishing time and distances

– Time can be found using impulse-momentum theorem, distance is found using work-KE theorem

• Larger velocities have larger KE• Larger changes in KE mean more work needs to be

done to stop the object• Work = Fd, more distance needed or more force needed

to stop faster moving objects• Ex. 2x’s speed means 4x stopping distance

Conservation of momentum

• Law of Conservation of p: total momentum of an isolated system is constant

• Momentum before must equal momentum after.

pai + pbi = paf + pbf

mavai + mbvbi = mavaf + mbvbf

• Where p is momentum, m is mass, and v is velocity. The subscripts a and b are for different objects involved and i and f are for initial and final.

• The total p of all objects interacting with one another remains constant regardless of nature of forces between objects

• Momentum is conserved in collisions

• Momentum is conserved for objects pushing away from each other

• Consider a recoil situation:• The momentum before firing is zero. After

firing, the net momentum is still zero because the momentum of the cannon is equal and opposite to the momentum of the cannonball.

For recoil or other situations where two objects are together initially such as• A bullet fired from a gun• A boy on a skateboard who passes a basketball• Two people on skateboards who push off of each

other• A cannonball fired from a cannon

• Initial mass is the sum of the two masses

(ma+ mb) vabi = mavaf + mbvbf

• Pay attention to signs of velocity!!

Practice Problem

• A 65.0 kg ice skater moving to the right with a velocity of 2.50 m/s throws a 0.150 kg snowball to the right with a velocity of 32.0 m/s relative to the ground.

• What is the velocity of the ice skater after throwing the snowball? Disregard the friction between the skates and ice

Collisions

Collision Types

• Elastic: two objects bounce apart after the collision so that they move separately

• Perfectly inelastic: two objects stick together after the collision so that their final velocities are the same

• Inelastic: two objects deform during the collision so that the total KE decreases, but the objects move separately after the collision

Inelastic Collisions

• In an inelastic collision between two freight cars, the momentum of the freight car on the left is shared with the freight car on the right.

• Assume perfectly inelastic collisions where the objects stick together

• Final mass is the sum of the two masses

mavai + mbvbi = (ma+ mb)vf

• Pay attention to signs of velocity!!

A cart moving at speed v collides with an identical stationary cart on an airtrack, and the two stick together after the collision. What is their velocity after colliding?a. vb. 0.5 vc. zerod. –0.5 ve. –vf. need more information

Consider the skater problem above

• A second skater initially at rest with a mass of 60.0 kg catches the snowball. What is the velocity of the second skater after catching the snowball in a perfectly inelastic collision?

Elastic Collisions

• When objects collide without being permanently deformed and without generating heat, the collision is an elastic collision and the total KE remains constant

• After the collision the two objects move separately • Colliding objects bounce perfectly in perfect elastic

collisions. • The sum of the momentum vectors is the same before

and after each collision.

mavai + mbvbi = mavaf + mbvbf

Elastic Collisions

a. A moving ball strikes a ball at rest.

b. Two moving balls collide head-on.

c. Two balls moving in the same direction collide.

Practice Problem

• Two billiard balls each with a mass of 0.35 kg strike each other head-on. One ball is initially moving left at 4.1 m/s and ends up moving right at 3.5 m/s. The second ball is initially moving to the right at 3.5 m/s.

• Find the final velocity of the 2nd ball

• You tee up a golf ball and drive it down the fairway. Assume that the collision of the golf club and ball is elastic. When the ball leaves the tee, how does its speed compare to the speed of the golf club?

a. Greater Than

b. Less than

c. Equal to