AP Physics

Impulse and Momentum

Which do you think has more momentum?

Change in momentumLike any change, change in momentum is

calculated by looking at final and initial momentums.

p = pf – pi

p: change in momentum pf: final momentum pi: initial momentum

Impulse = ∆MomentumConsider Newton’s 2nd

Law and the definition of acceleration

Units of Impulse: Units of Momentum:

Momentum is defined as “Inertia in Motion”

NsKg x m/s

Impulse is the AreaSince J=Ft, Impulse is the AREA of a Force vs.

Time graph.

Impulsive ForcesUsually high

magnitude, short duration.

Suppose the ball hits the bat at 90 mph and leaves the bat at 90 mph, what is the magnitude of the momentum change?

What is the change in the magnitude of the momentum?

ExampleA 100 g ball is dropped from a height of h = 2.00 m above the floor. It

rebounds vertically to a height of h'= 1.50 m after colliding with the floor. (a) Find the momentum of the ball immediately before it collides with the floor and immediately after it rebounds, (b) Determine the average force exerted by the floor on the ball. Assume that the time interval of the collision is 0.01 seconds.

How about a collision?Consider 2 objects

speeding toward each other. When they collide......

Due to Newton’s 3rd Law the FORCE they exert on each other are EQUAL and OPPOSITE.

The TIMES of impact are also equal.

Therefore, the IMPULSES of the 2 objects colliding are also EQUAL

21

21

2121

)()(

JJ

FtFt

ttFF

How about a collision?If the Impulses are equal then

the change in MOMENTUMS are also equal!

Momentum is conserved!The Law of Conservation of Momentum: “In the

absence of an external force (gravity, friction), the total momentum before the collision is equal to the total momentum after the collision.”

Law of Conservation of MomentumIf the resultant external force on a system

is zero, then the vector sum of the momentums of the objects will remain constant.

Pbefore = Pafter

Sample problem

A 75-kg man sits in the back of a 120-kg canoe that is at rest in a still pond. If the man begins to move forward in the canoe at 0.50 m/s relative to the shore, what happens to the canoe?

Collision TypesElastic collisions

Also called “hard” collisionsNo deformation occurs, no kinetic energy lost

Inelastic collisionsDeformation occurs, kinetic energy is lost

Perfectly Inelastic (stick together)Objects stick together and become one

objectDeformation occurs, kinetic energy is lost

Sample ProblemA fish moving at 2 m/s swallows a stationary fish which is 1/3 its mass. What is the velocity of the big fish after dinner?

Example Granny (m=80 kg) whizzes around the rink with a velocity of 6 m/s. She suddenly collides with Ambrose (m=40 kg) who is at rest directly in her path. Rather than knock him over, she picks him up and continues in motion without "braking." Determine the velocity of Granny and Ambrose.

Elastic CollisionIn elastic collisions, there is no

deformation of colliding objects, and no change in kinetic energy of the system. Therefore, two basic equations must hold for all elastic collisions

pb = pa (momentum conservation)

Kb = Ka (kinetic energy conservation)

What happens if we have two unknowns?At an amusement park, a 96.0 kg bumper

car moving with a speed of 1.24 m/s bounces elastically off a 135 kg bumper car at rest. Find the final velocities of the cars.

1 1 2 2 1 1f 2 2fm v m v m v m v

2 2 2 21 1 2 2 1 1f 2 2f

1 1 1 1.

2 2 2 2m v m v m v m v

1 1 1f

2 2f 2

( )1.

( )

m v v

m v v

Let subscript 1 refer to the 96.0kg car and subscript 2 refer to the 135 kg car.Use momentum conservation.

Use conservation of kinetic energy.

Rearranging the first equation gives

2 21 1 1f 1 1 1f 1 1f

2 22 2f 2 2f 22 2f 2

( ) ( )( )1 .

( )( )( )

m v v m v v v v

m v v v vm v v

1 1f

2f 2

( )1,

( )

v v

v v

2f 1 1f 2or .v v v v

Rearranging the second equation gives

Comparing these two equations implies that

2fv 1f .v

1 1 2 2 1 1f 2 1 1f 2

1 2 1 2 2 1 2 1f

( )( ) 2 ( )m v m v m v m v v vm m v m v m m v

1 2 21f 1 2

1 2 1 2

2m m mv v v

m m m m

2f 1 1f 2 1f 2f 2 1Since , .v v v v v v v v

Substitute for in the first equation and solve for

1fv 2f .v

1 1 2 2 1 2f 2 1 2 2f

1 1 2 1 2 1 2 2f

( )2 ( ) ( )m v m v m v v v m vm v m m v m m v

Substitute for in the first equation and solve for

Sample ProblemA 500-g cart moving at 2.0 m/s on an air track

elastically strikes a 1,000-g cart at rest. What are the resulting velocities of the two carts?

2D Inelastic Collisions must rely on the Conservation of Momentum:

Example: A car with a mass of 950 kg and a speed of 16 m/s approaches an intersection, as shown. A 1300 kg minivan traveling at 21 m/s is heading for the same intersection. The car and minivan collide and stick together. Find the speed and direction of the wrecked vehicles just after the collision, assuming external forces can be ignored.

Sample problem

Sample problemCalculate velocity of 8-kg ball after the collision.

3 m/s

2 kg 8 kg

0 m/s

Before

2 m/s

2 kg

8 kgv

After

50o

x

y

x

y