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CHAPTER 6

- Momentum
- and
- Collisions

Linear Momentum

- Newton formulated his version of dynamics based

on the concept of momentum (p), which he defined

as the product of the mass and velocity - Momentum Mass ? Velocity
- p mv
- Because velocity is a vector quantity, so to is

momentum. - The direction of the momentum is in the direction

of the velocity. - Units kg?m/s

Linear Momentum

- Example
- Rich Gossage set a fastball record by hurling a

0.14 kg baseball at a speed of 153 ft/sec. What

was the magnitude of the balls momentum as it

left his hand?

Linear Momentum

- Example
- A 0.149 baseball traveling at 28 m/s due south

approaches a waiting batter. The ball is hit and

momentarily crushed it springs back, sailing

away at 46 m/s due north. Determine the

magnitudes of it initial and final momenta and

the change in its momentum.

Impulse and Momentum Change

- When a force is applied to a body, there will be

a resulting proportional change in its motion. - Newtons Second Law states that the net force

applied to an object equals the resulting change

in its momentum per unit time. - Over a finite time interval, during which the

force may change, Newtons Second Law is - Fav ?p/?t

Impulse and Momentum Change

- Example
- A rocket fires its engine, which exerts an

average force of 1000 N for 40 seconds in a fixed

direction. What is the magnitude of the rockets

momentum change?

Impulse and Momentum Change

- It is useful to combine the force and time into a

single notion that equals the known momentum

change. - That single concept is called is called the

impulse of the force, and an alternative

statement of the Second Law becomes - Fav?t ?p
- A given impulse will produce a specific change in

momentum no matter what the mass or speed of the

recipient body.

Impulse and Momentum Change

- If an object is originally at rest, it will take

off in the direction of the net applied force,

acquiring a momentum. - Force applied to a body already in motion may

either increase or decrease its momentum

depending on whether the force acts parallel or

antiparallel to the initial velocity. - Example In order to turn a pitched baseball

arriving at 90 mph into a homerun leaving at 110

mph, a bat must apply a force of up to 8000 lbs.

During this impact, which lasts only 1.25 ms,

the ball is crushed to half of its diameter.

Impulse and Momentum Change

- Example
- On September 12, 1966, a Gemini spacecraft

piloted by astronauts Pete Conrad and Richard

Gordon met and docked with an orbiting Agena

launch vehicle. With plenty of fuel left in the

spacecraft, NASA decided to determine the mass of

the Agena. While coupled, Geminis motor was

fired, exerting a constant thrust of 890 N in a

fixed direction for 7.0 s. As a result of that

little nudge, the Gemini-Agena sped up by 0.93

m/s. Assuming Geminis mass as a constant 34 ?

102 kg, compute the mass of the Agena.

Varying Force

- Force may change from moment to moment.
- It might be easier to represent force by a curve

(see diagram). - The force starts out small as the object comes in

contact with another object, rises to a maximum

value when they are firmly in contact, and then

drops off as the objects lose contact with each

other.

Varying Force

- Just as the area under any portion of the speed

time curve is the distance traveled during a

given period, the area under the force-time curve

is the impulse exerted during that time interval

and it therefore equals the resulting change in

momentum. - Area ½?tFmax
- ?p ½?tFmax

Varying Force

- A real force-time curve can have a complex shape.
- Example A spaceship with a variable-thrust

engine can generate a force-time curve with many

bumps and wiggles, but the area encompassed in a

given time interval will always be equal to the

resulting change in momentum. - When you throw a punch, your fist starts with pi

0 and ends with pf 0. It accelerates up to a

maximum speed and then decelerates to zero when

your arm is extended. - Given this fact, a karate blow is always aimed at

a point inside the target so that it makes

contact when p is maximum.

Varying Force

- Example
- A golfers club hits a 47.0 g golf ball from rest

to a speed of 60.0 m/s in a collision lasting

1.00 ms. The force on the ball rises to a peak

value of Fmax and then drops to zero as it leaves

the club. Compute a rough value for this maximum

force by approximating the force-time curve, with

a triangle of altitude Fmax.

Car Crashes

- When a car crashes into a brick wall, its front

end deforms as it slows to a crushing stop. - On average, automobiles compress roughly 1 inch

for every mile per hour of speed just prior to

impact. - If we assume that the collision-deceleration is

fairly uniform, the crush distance (sc) divided

by vav ½(vi vf), with vf 0, is the impact

time, thus - ?t sc/vav
- ?t 2sc/vi

Car Crashes

- Since tests show that sc is proportional to vi,

the impact time for cars of comparable stiffness

should be independent of the speed of impact. - A typical head-on, brick-wall collision lasts

around 100 ms (airbags inflate in 55 ms). - Crashing head-on into an identical car traveling

toward you at the same speed is effectively the

same as hitting a stationary brick wall.

Car Crashes

- Example
- A 70-kg passenger riding in a typical automobile

is involved in a 40 mph head-on collision with a

concrete barrier. Taking the stopping time as 100

ms, compute the average force exerted by the

seatbelt and shoulder strap on the person.

Jets and Rockets

- Imagine yourself on roller skates holding a bag

of oranges. - Now you throw one of them due north, and away you

go due south. Why? - You push on the orange in the forward direction

during the throw, it pushes back with the same

impulse on you, and back you go. - A rocket works by hurling out a tremendous number

of tiny high-speed (3 to 4-km/s) objects

(molecules). - Example During launch, the solid fuel boosters

for the Space Shuttle expel 8.5 tons of fiery

exhaust each second. The engines blast exhaust

downward, and the escaping gas, in turn, pushes

the up on the object (action/reaction).

Jets and Rockets

- Example
- A rocket engine testing a low-power fuel expels

5.0 kg of exhaust gas per second. If these

molecules are ejected at an average speed of 1.2

km/s, what is the thrust of the engine?

Conservation of Linear Momentum

- Conservation of Momentum
- When the resultant of all the external forces

acting on a system is zero, the linear momentum

of the system remains constant (pi pf). - The net momentum of an isolated system cannot

change. - The total momentum of a system of interacting

masses must remain unaltered, provided that no

net external force is applied.

Conservation of Linear Momentum

- Example
- According to published figures, a bullet fired

from a standard 9-mm Luger pistol has a mass of

8.0 grams and a muzzle speed of 352 m/s. If the

mass of the gun is 0.90 kg, what is the recoil

speed when fired horizontally?

Conservation of Linear Momentum

- Since neither the gun nor bullet were moving

prior to firing the pistol, they have no initial

momentum. - pi pf
- pi 0 pf
- 0 mbvbf mgvgf
- 0 (0.0080 kg)(352 m/s) (0.90 kg)vgf
- vgf -(0.0080 kg)/(0.90 kg) ? 352 m/s
- vgf -3.1 m/s
- The (-) sign indicates that the gun is moving in

the opposite direction of the bullet.

Conservation of Linear Momentum

- Example
- While floating in space a 100-kg robot throws a

0.800 kg wrench at 12.0 m/s toward his partner

working on the spaceship. How fast will the robot

move away from the ship? - pi pf
- 0 mrvrf mwvwf
- 0 (100 kg)vrf (0.800 kg)(12.0 m/s)
- vrf -0.096 m/s
- vrf -0.1 m/s

Collisions

- A collision is marked by the transfer of momentum

between objects in relative motion resulting from

their interaction via at least one of the four

forces. - In all cases where there are no external forces

acting, the total momentum of the colliding

objects is conserved. - There are two types of collisions that we will be

discussing in this chapter.

Collisions

- Inelastic Collisions
- An inelastic collision is one where the final

kinetic energy (KE ½mv2) of the system is

different from the initial KE. - KEi ? KEf
- Example
- When you drop a tennis ball to the floor, it

momentarily comes to rest, and then springs back,

popping into the air. But the squashing of the

ball produces some internal heating, and the ball

only returns about two-thirds of the way back up. - All collisions between macroscopic objects are

more or less inelastic. - The completely inelastic collision is at one

extreme where the impacting objects stick

together and the maximum amount of KE is

transformed (i.e. lost) into internal energy. - m1v1i m2v2i (m1 m2)vf

Collisions

- Example
- During a rainy day football game, a 854 N

quarterback is standing holding the ball looking

for a receiver when hes unkindly hit by a 1281 N

tackle charging in at 6.1 m/s. - At what speed do the two men, tangled together,

initially sail off on the wet field? (Disregard

friciton) - How much mechanical energy is lost to friction?

Collisions

- pi pf
- mqvqi mtvti (mq mt)vf
- 0 (1281/9.81)(6.1) ((854/9.81)

(1281/9.81))vf - 796.8 217.7vf
- vf 3.7 m/s
- KEi ½mqvq2 ½ mtvt2
- KEi 0 ½(1281/9.81)(6.1)2
- KEi 2.4 kJ
- KEf ½(mq mt)vf2
- KEf ½((854/9.81) (1281/9.81))(3.7)2
- KEf 1.5 kJ
- So, 2.4 1.5 0.9 kJ of energy was lost.

Collisions

- For completely inelastic collisions between two

bodies, only one of which (m1) is moving, the

ratio of the total final KE to the total initial

KE is - KEf (m1/(m1 m2))KEi

Collisions

- 2. Elastic Collisions
- A collision is elastic when KE is constant.
- KEi KEf
- KE1i KE2i KE1f KE2f
- The relative speeds of the two bodies before

and after an elastic impact are equal. - The relative velocity is reversed by the

collision, but its magnitude is unchanged. - This is true for all elastic collisions.
- Example Newtons Cradle.

Collisions

- Example
- Two billiard balls move toward one another. The

balls have identical masses, and assume that the

collision between them is perfectly elastic. If

the initial velocities of the balls are 30.0

cm/s and -20.0 cm/s, what is the velocity of each

ball after the collision.?

Collisions in Two Dimensions

- When two objects with masses m1 and m2

elastically slam into one another off center

(glancing collision) they subsequently move away

at different angles to the original direction. - Example Playing pool.
- Provided that there are no external forces

acting, momentum is always conserved. - Because of the 2-D of the collisions, we will

make use of the fact that the scalar momentum

components in the x and y directions are

conserved independently. - pix pfx and piy pfy

Collisions in Two Dimensions

- In a glancing collision, when the two objects

have an equal mass (m1 m2), and one of them is

at rest, the two will always move off at right

angles to each other. - We will only be dealing with objects in glancing

collisions that stick together after the

collision in order to avoid having too many

unknowns.

Collisions in Two Dimensions

- Example
- Two cars enter an icy intersection and skid into

each other. The 2.50 103 kg sedan was

originally heading south at 20.0 m/s, whereas the

1.45 103 kg coupe was driving east at 30.0 m/s.

On impact, the two vehicles become entangled and

move off as one at an angle (?) in a

southeasterly direction. Determine the angle and

the speed at which they initially skid away after

crashing.

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