Momentum

• A measure of how difficult it is to change an object’s motion (to make it stop or swerve).

• On what does this difficulty depend?–More mass; more difficulty–More velocity; more difficulty

• Define Momentum: p = mv; a vector quantity

Momentum = mass × velocity

of object of object of object

Impulse

• How do we change an object’s momentum?– Apply a net force for a period of time– Larger net force; more momentum– Longer time; more momentum

• Let’s define an Impulse:

• JNET = FNET Δt; a vector quantity

• When we apply a net force to an object over a period of time, we are applying an impulse to the object:

Impulse applied = Net Force × time during which net

force is applied.

How is the Impulse applied to an object related to its change in momentum?

• JNET = FNET Δt (by definition of JNET) = (ma) Δt (by the Second Law) = (m Δv/ Δt ) Δt (by definition of a=Δv/Δt ) = m Δv = Δp = Change in Momentum

• So: Net Impulse applied to an object equals The Object’s change in momentum• FNET Δt = m Δv ; JNET = Δp (alternate version of the 2nd law)

How can I stop a moving car’s momentum with an impulse?

• Two ways to stop a car initially moving at 60 miles/hour• Apply the Brake:

– Applies a small force for a long period of time.

– FBRAKE (small) × Δt (large)

• Hit a Brick Wall:– Applies a large force for a short period of time.

– FWALL(large) × Δt (small)

• What is the change in momentum for each case?

• Same Δp and same JNET but the brake is safer.

• How do airbags work to make a car collision safer?

Units and Some Numbers

• Do JNET and Δp have the same units?– S.I. units are kg·m/s (there is no special defined unit)

Example: A car with a mass: m=1000kg moves with a velocity of 10m/s.

• What is its momentum?

• What impulse is needed to bring it to a halt?

• What braking force is needed to bring the car to a stop in 5s?

Sticking and Bouncing

• A man throws a 2 kg ball at the wall with a velocity of 6 m/s East. The ball sticks to the wall.– What is the initial velocity (vi)?– What is the final velocity (vf)?– What is the change in momentum(Δp = m Δv = m(vf - vi))?– What is the impulse delivered by the wall (JNET)?

• The man throws a similar ball at the same velocity at the wall but this ball bounces off at 6m/s West.– What is the initial velocity (vi)?– What is the final velocity (vf)?– What is the change in momentum(Δp = m Δv = m(vf - vi))?– What is the impulse delivered by the wall (JNET)?

• In which case is the impulse delivered greater? Bouncing or Sticking?

Total Momentum of a System of Objects

• A “System” is an object or a collection of objects.

• The Total Momentum of a system equals the vector sum of the momenta of all the objects in the system:

• PTotal System = P1 + P2 (for a system of two objects)

• Also called the “Net Momentum”: PNET or PTOT

• EXAMPLES:– We have two cars, each with mass: m=6 kg. One travels

at 20m/s East and the other at 20m/s West. What is PTOT?

– What is PTOT if both cars travel at 20 m/s East?

Conservation of Momentum

• Conservation of Momentum of a System– If no net external force or impulse acts on a

system, then its momentum remains unchanged.

• Example: A 1000 kg cannon fires a 20 kg ball at a velocity of 20 m/s East. With what velocity does the cannon recoil backward?

(Hint: The firing of the cannon is an internal force)

Recipe for Solving Momentum Conservation Problems

1. Draw a initial and final pictures of the problem. Write down: knowns and unknowns.

2. Write an expression for the total initial momentum of the system. (pTOTi = <initial pTOT> )

3. Write an expression for the total final momentum of the system (pTOTf = <final pTOT> )

4. Set the expression for the total initial momentum equal to the expression for the total final momentum: <initial pTOT > = <final pTOT>

5. Solve for unknowns.

Collisions

• In any collision between objects (with no external forces applied), the net (or total) momentum before collision equals the net momentum after collision. PTOT (before) = PTOT (after)

• Elastic Collision: – Frictionless, no deformation, no heating– Kinetic Energy is conserved

• (K.E. = ½ mv2 we’ll use this later)

• Inelastic Collision:– Friction, heating, deformation, momentum is shared– Kinetic Energy is lost to heat and deformation.

Elastic Collisions

Inelastic Collisions

An Inelastic Collision Problem• A cart of mass 40 kg traveling at 20 m/s hits a

stationary cart of mass 60 kg and the two couple together. How fast do the two cars (traveling as one) move?

Another Inelastic Collision Problem

Suppose the small fish was swimming towards the big fish at 6 m/s? What would be the big fish’s speed and direction after lunch?