Momentum and Momentum and CollisionsCollisions

Linear Momentum

The linear momentum of a particle or an object that can be modeled as a particle of mass m moving with a velocity v is defined to be the product of the mass and velocity: p = m v

We will usually refer to this as “momentum”, omitting the “linear”.

Momentum is a VECTOR. It has components. Don’t forget that.

Linear Momentum

The dimensions of momentum are ML/T The SI units of momentum are kg · m / s Momentum can be expressed in component

form: px = m vx py = m vy pz = m vz

A 3.00-kg particle has a velocity of . (a) Find its x and y components of momentum. (b) Find the magnitude and direction of its momentum.

Newton and Momentum

Newton called the product mv the quantity of motion of the particle

Newton’s Second Law can be used to relate the momentum of a particle to the resultant force acting on it

with constant mass.

d md dm m

dt dt dt

vv pF a

Newton’s Second Law

The time rate of change of the linear momentum of a particle is equal to the net force acting on the particle This is the form in which Newton presented the Second

Law It is a more general form than the one we used previously This form also allows for mass changes

Applications to systems of particles are particularly powerful

Two Particles (1 dimension for now)

other. on the force

a producing particle one are that those

are system on this acting forcesonly the

thenforces external NO are there

21

IFdt

dp

dt

dp

dt

dptotal

Continuing

00 int

21

ernaltotal

external

total

Fdt

dpF

dt

dp

dt

dp

dt

dp

NO EXTERNAL FORCES: ptotal is constant Force on P1 is from particle 2 and is F21

Force on P2 is from particle 1 and is F12

F21=-F12

Summary

The momentum of a SYSTEM of particles that areisolated from external forces remains a constant ofthe motion.

Conservation of Linear MomentumTextbook Statement Whenever two or more particles in an

isolated system interact, the total momentum of the system remains constant The momentum of the system is conserved,

not necessarily the momentum of an individual particle

This also tells us that the total momentum of an isolated system equals its initial momentum

Two blocks of masses M and 3M are placed on a horizontal, frictionless surface. A light spring is attached to one of them, and the blocks are pushed together with the spring between them). A cord initially holding the blocks together is burned; after this, the block of mass 3M moves to the right with a speed of 2.00 m/s. (a) What is the speed of the block of mass M? (b) Find the original elastic potential energy in the spring if M = 0.350 kg.

A particle of mass m moves with momentum p. Show that the kinetic energy of the particle is K = p2/2m. (b) Express the magnitude of the particle’s momentum in terms of its kinetic energy and mass.

An Easy One…

Consider a particle roaming around that is suddenly subjected to some kind of FORCE that looks something like the last slide’s graph.

change will then 0 If

0 ifconstant

:

pF

Fp

Favp

vp

mdt

dm

dt

d

m

NEWTON

Let’s drop the vector notation and stick to one dimension.

f

i

f

i

t

t

if

f

i

t

t

Fdtpp

Fdtdp

Fdtdp

Fdt

dp

Change of Momentum

Impulse

NEW LAW

IMPULSE = CHANGE IN MOMENTUM

NEW LAW

A 3.00-kg steel ball strikes a wall with a speed of 10.0 m/s at an angle of 60.0° with the surface. It bounces off with the same speed and angle (Fig. P9.9). If the ball is in contact with the wall for 0.200 s, what is the average force exerted on the ball by the wall?

LETS TALK ABOUT

Consider two particles:

1 2

m1 m2

v1v2

V1 V2

1 2

1 2

What goes on during the collision? N3

Force on m2

=F(12)

Force on m1

=F(21)

The Forces

Equal and Opposite

More About Impulse: F-t The Graph Impulse is a vector quantity The magnitude of the

impulse is equal to the area under the force-time curve

Dimensions of impulse are M L / T

Impulse is not a property of the particle, but a measure of the change in momentum of the particle

Impulse

The impulse can also be found by using the time averaged force

I = t

This would give the same impulse as the time-varying force does

F

SKATEBOARD DEMO

Conservation of Momentum, Archer Example

The archer is standing on a frictionless surface (ice)

Approaches: Newton’s Second Law –

no, no information about F or a

Energy approach – no, no information about work or energy

Momentum – yes

Archer Example, 2

Let the system be the archer with bow (particle 1) and the arrow (particle 2)

There are no external forces in the x-direction, so it is isolated in terms of momentum in the x-direction

Total momentum before releasing the arrow is 0 The total momentum after releasing the arrow is p1f

+ p2f = 0

Archer Example, final

The archer will move in the opposite direction of the arrow after the release Agrees with Newton’s Third Law

Because the archer is much more massive than the arrow, his acceleration and velocity will be much smaller than those of the arrow

An estimated force-time curve for a baseball struck by a bat is shown in Figure P9.7. From this curve, determine (a) the impulse delivered to the ball, (b) the average force exerted on the ball, and (c) the peak force exerted on the ball.

Overview: Collisions – Characteristics We use the term collision to represent an event

during which two particles come close to each other and interact by means of forces

The time interval during which the velocity changes from its initial to final values is assumed to be short

The interaction force is assumed to be much greater than any external forces present This means the impulse approximation can be used

Collisions – Example 1

Collisions may be the result of direct contact

The impulsive forces may vary in time in complicated ways This force is internal to

the system Momentum is

conserved

Collisions – Example 2

The collision need not include physical contact between the objects

There are still forces between the particles

This type of collision can be analyzed in the same way as those that include physical contact

Types of Collisions

In an elastic collision, momentum and kinetic energy are conserved Perfectly elastic collisions occur on a microscopic level In macroscopic collisions, only approximately elastic

collisions actually occur In an inelastic collision, kinetic energy is not

conserved although momentum is still conserved If the objects stick together after the collision, it is a

perfectly inelastic collision

Collisions, cont

In an inelastic collision, some kinetic energy is lost, but the objects do not stick together

Elastic and perfectly inelastic collisions are limiting cases, most actual collisions fall in between these two types

Momentum is conserved in Momentum is conserved in allall collisions collisions

Perfectly Inelastic Collisions

Since the objects stick together, they share the same velocity after the collision

m1v1i + m2v2i =

(m1 + m2) vf

A 10.0-g bullet is fired into a stationary block of wood (m = 5.00 kg). The relative motion of the bullet stops inside the block. The speed of the bullet-plus-wood combination immediately after the collision is 0.600 m/s. What was the original speed of the bullet?

. Velcro couplers make the carts stick together after colliding. Find the final velocity of the train of three carts. (b) What If? Does your answer require that all the carts collide and stick together at the same time? What if they collide in a different order?

Most of us know intuitively that in a head-on collision between a large dump truck and a subcompact car, you are better off being in the truck than in the car. Why is this? Many people imagine that the collision force exerted on the car is much greater than that experienced by the truck. To substantiate this view, they point out that the car is crushed, whereas the truck is only dented. This idea of unequal forces, of course, is false. Newton’s third law tells us that both objects experience forces of the same magnitude. The truck suffers less damage because it is made of stronger metal. But what about the two drivers? Do they experience the same forces? To answer this question, suppose that each vehicle is initially moving at 8.00 m/s and that they undergo a perfectly inelastic head-on collision. Each driver has mass 80.0 kg. Including the drivers, the total vehicle masses are 800 kg for the car and 4 000 kg for the truck. If the collision time is 0.120 s, what force does the seatbelt exert on each driver?

Elastic Collisions

Both momentum and kinetic energy are conserved

1 1 2 2

1 1 2 2

2 21 1 2 2

2 21 1 2 2

1 1

2 21 1

2 2

i i

f f

i i

f f

m m

m m

m m

m m

v v

v v

v v

v v

Elastic Collisions, cont Typically, there are two unknowns to solve for and so you need two

equations The kinetic energy equation can be difficult to use With some algebraic manipulation, a different equation can be used

v1i – v2i = v1f + v2f This equation, along with conservation of momentum, can be used

to solve for the two unknowns It can only be used with a one-dimensional, elastic collision

between two objects The solution is shown on pages 262-3 in the textbook. (Lots

of algebra but nothing all that difficult. We will look at a special case.

Let’s look at the case of equal masses.

Second Particle initially at rest

Explains the demo!

Elastic Collisions, final

Example of some special cases m1 = m2 – the particles exchange velocities When a very heavy particle collides head-on with a very

light one initially at rest, the heavy particle continues in motion unaltered and the light particle rebounds with a speed of about twice the initial speed of the heavy particle

When a very light particle collides head-on with a very heavy particle initially at rest, the light particle has its velocity reversed and the heavy particle remains approximately at rest

Collision Example – Ballistic Pendulum Perfectly inelastic collision –

the bullet is embedded in the block of wood

Momentum equation will have two unknowns

Use conservation of energy from the pendulum to find the velocity just after the collision

Then you can find the speed of the bullet