chapter 7 momentum and collisions. momentum newton’s laws give a description of forces ○ there...
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Chapter 7Momentum and Collisions
MomentumNewton’s Laws give a description of forces
○ There is a force acting or their isn’t○ But what about in between
Constant velocity
Momentum is a quantity defined by the velocity of a mass○ Velocity changes -> Force
Impulse
Momentum describes collisions○ Magnitude gives an indication of how hard it is to
stop○
Momentum From Newton’s laws: force must be present to change an object’s
velocity (speed and/or direction) Wish to consider effects of collisions and corresponding change
in velocity
Method to describe is to use concept of linear momentum
scalar vector
Linear momentum = product of mass velocityLinear momentum = product of mass velocity
Golf ball initially at rest, so some of the KE of club transferred to provide motion of golf ball and its change in velocity
Momentum and Impulse
The linear momentum p of an object of mass m moving with velocity v is defined as
p = mv
Same direction as velocityUnits
○ kg x m / s
m v
p
Momentum
Vector quantity, the direction of the momentum is the same as the velocity’s
Applies to two-dimensional motion as well
yyxx m vpa n dm vp
vmp
Size of momentum: depends upon mass depends upon velocity
Impulse
In order to change the momentum of an object (say, golf ball), a force must be applied
The time rate of change of momentum of an object is equal to the net force acting on it
Gives an alternative statement of Newton’s second law(F Δt) is defined as the impulseImpulse is a vector quantity, the direction is the same as the
direction of the force
tFporamt
vvm
t
pF n et
ifn et
:)(
Graphical Interpretation of Impulse
Usually force is not constant, but time-dependent
If the force is not constant, use the average force applied
The average force can be thought of as the constant force that would give the same impulse to the object in the time interval as the actual time-varying force gives in the interval
( )i
i it
im pu lse F t a rea under F t cu rve
If force is constant: impulse = F t
Example: Impulse Applied to Auto Collisions The most important factor is the collision time
or the time it takes the person to come to a restThis will reduce the chance of dying in a car crash
Ways to increase the timeSeat beltsAir bags
The air bag increases the The air bag increases the time of the collisiontime of the collision and and absorbs some of the energyabsorbs some of the energy from the from the bodybody
ConcepTest
Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the time intervals to stop them compare?
1. It takes less time to stop the ping-pong ball.2. Both take the same time.3. It takes more time to stop the ping-pong ball.
ConcepTest
Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the time intervals to stop them compare?
1. It takes less time to stop the ping-pong ball.2. Both take the same time.3. It takes more time to stop the ping-pong ball.
Note: Because force equals the time rate of change of momentum, the two balls loose momentum at the same rate. If both balls initially had the same momenta, it takes the same amount of time to stop them.
Problem: Teeing Off
A 50-g golf ball at rest is hit by “Big Bertha” club with 500-g mass. After the collision, golf leaves with velocity of 50 m/s.
a) Find impulse imparted to ballb) Assuming club in contact with
ball for 0.5 ms, find average force acting on golf ball
Problem: teeing off
Given:
mass: m=50 g = 0.050 kgvelocity: v=50 m/s
Find:
impulse=?Faverage=?
1. Use impulse-momentum relation:
2. Having found impulse, find the average force from the definition of impulse:
smkg
smkg
mvmvpimpulse if
50.2
050050.0
N
s
smkg
t
pFthustFp
3
3
1000.5
105.0
50.2,
Note: according to Newton’s 3rd law, that is also a reaction force to club hitting the ball:
Momentum and Impulse
Can describe Newton’s laws more precisely with momentum
Law I : If no forces act, Fnet = 0 so a = 0 which implies that velocity is constant p = mv, therefore momentum is constantMomentum constant => conserved quantity
Law II : F = Δp / Δt = m (Δv / Δt) = ma Law III : Rate of change of momentum
generated by action force is exactly opposite to the rate of change of momentum by the reaction force
Conservation of Momentum Definition: an isolated system is the one that
has no external forces acting on it
A collision may be the result of physical contact between two objects
“Contact” may also arise from the electrostatic interactions of the electrons in the surface atoms of the bodies
Momentum in an isolated system in Momentum in an isolated system in which a which a collision occurs is conserved (regardless collision occurs is conserved (regardless of the nature of the forces between the of the nature of the forces between the objects)objects)
Conservation of Momentum
The principle of conservation of momentum states when no external forces act on a system consisting of two objects that collide with each other, the total momentum of the system before the collision is equal to the total momentum of the system after the collision
Conservation of Momentum
Mathematically:
Momentum is conserved for the system of objectsThe system includes all the objects interacting with each otherAssumes only internal forces are acting during the collisionCan be generalized to any number of objects
ffii vmvmvmvm 22112211
Momentum and Impulse
Example:
Conservation of Momentum If the isolated system consists of two
objects undergoing a collision, the total momentum of the system is the same before and after the collision. Same concept as energy
m1v1i + m2v2i = m1v1f + m2v2f
Conservation of Momentum Remember conservation of momentum
applies to the systemThe system includes all the objects
interacting with each otherAssumes only internal forces are acting
during the collision
You must define the isolated systemCan be generalized to any number of
objects
ConcepTest 7.1 Rolling in the Rain
1) speeds up1) speeds up
2) maintains constant speed2) maintains constant speed
3) slows down3) slows down
4) stops immediately4) stops immediately
An open cart rolls along a
frictionless track while it is
raining. As it rolls, what happens
to the speed of the cart as the rain
collects in it? (assume that the
rain falls vertically into the box)
Since the rain falls in vertically, it adds
no momentum to the box, thus the box’s
momentum is conserved. However,
since the mass of the box slowly
increasesincreases with the added rain, its
velocity has to decreasedecrease.
ConcepTest 7.1 Rolling in the Rain
1) speeds up1) speeds up
2) maintains constant speed2) maintains constant speed
3) slows down3) slows down
4) stops immediately4) stops immediately
An open cart rolls along a
frictionless track while it is
raining. As it rolls, what happens
to the speed of the cart as the rain
collects in it? (assume that the
rain falls vertically into the box)
Follow-up:Follow-up: What happens to the cart when it stops raining? What happens to the cart when it stops raining?
Collisions
Momentum is conserved in any collision We can define two types of collisions to
describe how momentum is transferred within the systemElastic
○ Both momentum and kinetic energy are conserved
Inelastic○ Kinetic energy is not conserved, momentum is
conserved
Collisions
Elastic Collisions Momentum is conserved Total Energy is
conserved
2f22
2f11
2i22
2i11
f22f11i22i11
vm2
1vm
2
1vm
2
1vm
2
1
vmvmvmvm
Collisions
Elastic CollisionsInstead of using the energy quadratic
equation, we can simplify the equation to
The relative velocity of the two objects before the collision equals the negative of the relative velocity of the two objects after the collision
)vv(vv f2f1i2i1
Collisions
Elastic CollisionConsider the case: both targets movingSo, m1v1i + m2v2i = m1v1f + m2v2f
½m1v21i + ½ m2v2
2i = ½ m1v21f + ½m2v2
We know m1, m2, v1i, and v2i
Solve for v1f, v2f
Collisions
General Equation
v1f = ————— v1i + ————— v2i
v2f = ————— v1i + ————— v2i
m1 – m2 2m2
2m2 m1 – m2
m1 +m2 m1 + m2
m1 +m2 m1 + m2
Collisions
Consider the case v2i = 0
we get,
v1f = ———— v1i
v2f = ———— v1i
m1 – m2
2m2
m1 + m2
m1 + m2
Collisions (special cases)
Special situations
1. Equal masses (m2 = m1) Exchange velocities
If v2i = 0, v1f = 0 and v2f = v2i
○ body 1 stops and body 2 takes off with speed v1i
Collisions (special cases)
1. Massive target (m2 >> m1) v1f ≈ -v1i and v2f ≈ (2m1 / m2)v1i
body 1 bounces back in opposite direction, body 2 moves slowly
○ Example: golf ball at cannonball
Collisions (special cases)
1. Massive projectile (m1 >> m2) v1f ≈ v1i and v2f ≈ 2v1i
Body 1 continues at same speed, body two at twice the speed!
○ Example: cannonball at golf ball
Collisions Example
Types of Collisions
Momentum is conserved in any collision
what about kinetic energy?
Inelastic collisions Kinetic energy is not conserved
○ Some of the kinetic energy is converted into other types of energy such as heat, sound, work to permanently deform an object
Perfectly inelastic collisions occur when the objects stick together○ Not all of the KE is necessarily lost
Actual collisions Most collisions fall between elastic and perfectly inelastic
collisions
e n e r g ylo s t fi K EK E
Collisions
Perfectly InelasticObjects stick together
○ Not all of the KE is necessarily lost
Most real life collisions fall between elastic and perfectly inelastic collisions
Collisions
Perfectly Inelastic CollisionsSince both masses
have the same velocity, conservation of momentum becomes,
m1v1i + m2v2i = (m1 + m2)vf
Collisions Example of perfectly inelastic collision
Momentum is conserved, kinetic energy is not conserved and converted into heat, sound, and destruction of smaller car
Collisions
Application – Perfectly Inelastic CollisionBallistic Pendulum
○ Initially used to measure speeds of bullets before electronic timing devices were developed.
Sketches for Collision Problems Draw “before” and
“after” sketches Label each object
include the direction of velocity
keep track of subscripts
Sketches for Perfectly Inelastic Collisions The objects stick
together Include all the
velocity directions The “after” collision
combines the masses
Perfectly Inelastic Collisions:
When two objects stick together after the collision, they have undergone a perfectly inelastic collision
Suppose, for example, v2i=0. Conservation of momentum becomes
fii vmmvmvm )( 212211
.20105.2
105
,)2500(0)50)(1000(
:1500,1000ifE.g.,
3
4
21
smkg
smkgv
vkgsmkg
kgmkgm
f
f
fi vmmvm )(0 2111
Perfectly Inelastic Collisions: What amount of KE lost
during collision?
Jsmkg
vmvmKE iibefore
62
222
211
1025.1)50)(1000(2
12
1
2
1
Jsmkg
vmmKE fafter
62
221
1050.0)20)(2500(2
1
)(2
1
JK E l o s t61 07 5.0
lost in heat/”gluing”/sound/…
Collisions Example – Perfectly Inelastic Collision
Two objects, one with mass of 5kg and the other 3kg, approach each other with velocities of 2m/s from opposite directions. After they collide, they stick together. What is the initial and final kinetic energy?
M = 5 kg M = 3 kg
Glancing Collisions
For a general collision of two objects in three-dimensional space, the conservation of momentum principle implies that the total momentum of the system in each direction is conserved
Use subscripts for identifying the object, initial and final, and components
fyfyiyiy
fxfxixix
vmvmvmvm
andvmvmvmvm
22112211
22112211
Glancing Collisions
The “after” velocities have x and y components Momentum is conserved in the x direction and in
the y direction Apply separately to each direction
Two-dimensional Collisions
For a general collision of two objects in three-dimensional space, the conservation of momentum principle
… implies that the total momentum of the system in each direction is conserved
Use subscripts for identifying the object, initial and final, and components
fyfyiyiy
fxfxixix
vmvmvmvm
vmvmvmvm
22112211
22112211 and
ffii vmvmvmvm 22112211
Glancing Collisions
Two dimensional collisionsMomentum is still conservedApply vector components to conservation of
momentum equation
ApplicationsBilliard ballsParticle physics
Center of MassIn (a), the diver’s motion is pure translation; in (b) it is translation plus rotation.
There is one point that moves in the same path a particle would take if subjected to the same force as the diver. This point is called the center of mass (CM).
Center of Mass
The general motion of an object can be considered as the sum of the translational motion of the CM, plus rotational, vibrational, or other forms of motion about the CM.
Center of Mass
For two particles, the center of mass lies closer to the one with the most mass
where M is the total mass.
Center of Mass
Example