Physics 151: Lecture 18, Pg 1 Physics 151: Lecture 18 Today’s Agenda l Topics çReview of momentum...
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Transcript of Physics 151: Lecture 18, Pg 1 Physics 151: Lecture 18 Today’s Agenda l Topics çReview of momentum...
Physics 151: Lecture 18, Pg 1
Physics 151: Lecture 18Physics 151: Lecture 18
Today’s AgendaToday’s Agenda
TopicsReview of momentum conservation2-D CollisionsSystems of particles
Physics 151: Lecture 18, Pg 2
Lecture 17 Lecture 17 ACT 4ACT 4
The law of conservation of momentum applies to a collision between two bodies if:
a. they exert forces on each other respectively proportional to their masses.
b. they exert forces on each other respectively proportional to their velocities.
d. their accelerations are proportional to their masses.
e. they exert equal and opposite forces on each other.
Physics 151: Lecture 18, Pg 3
Lecture 18Lecture 18Review problem: Review problem: qualitative qualitative
Two boys in a canoe toss a baseball back and forth. What effect will this have on the canoe? Neglect (velocity-dependent) frictional forces with water or air.
a. None, because the ball remains in the canoe.
b. The canoe will drift in the direction of the boy who throws the ball harder each time.
c. The canoe will drift in the direction of the boy who throws the ball with less force each time.
d. The canoe will oscillate back and forth always moving opposite to the ball.
e. The canoe will oscillate in the direction of the ball because the canoe and ball exert forces in opposite directions upon the person throwing the ball.
Physics 151: Lecture 18, Pg 4
Lecture 17, Lecture 17, ACT 3ACT 3Momentum ConservationMomentum Conservation
Two balls of equal mass are thrown horizontally with the same initial velocity. They hit identical stationary boxes resting on a frictionless horizontal surface.
The ball hitting box 1 bounces back, while the ball hitting box 2 gets stuck.Which box ends up moving fastest ?
(a)(a) Box 1 (b)(b) Box 2 (c)(c) same
1 2
Physics 151: Lecture 18, Pg 5
Lecture 17Lecture 17Review problemReview problem
A 3.0-kg mass is sliding on a horizontal frictionless surface with a speed of 3.0 m/s when it collides with a 1.0-kg mass initially at rest as shown in the figure. The masses stick together and slide up a frictionless circular track of radius 0.40 m. To what maximum height, h, above the horizontal surface will the masses slide?
Physics 151: Lecture 18, Pg 6
Inelastic collision in 2-DInelastic collision in 2-D
Consider a collision in 2-D (cars crashing at a slippery intersection...no friction).
vv1
vv2
VV
before after
m1
m2
m1 + m2
Physics 151: Lecture 18, Pg 7
Inelastic collision in 2-D...Inelastic collision in 2-D...
There are no net external forces acting. Use momentum conservation for both components.
vv1
vv2
VV = (Vx,Vy)
m1
m2
m1 + m2
b,xa,x PP m v m m Vx1 1 1 2
Vm
m mvx
1
1 21
b,ya,y PP m v m m Vy2 2 1 2
Vm
m mvy
2
1 22
Physics 151: Lecture 18, Pg 8
Inelastic collision in 2-D...Inelastic collision in 2-D...
So we know all about the motion after the collision !
VV = (Vx,Vy)
Vx
Vy
V
m
m mvx
1
1 21
V
m
m mvy
2
1 22
1
2
11
22
x
y
pp
vmvm
V
Vtan PP
pp1
pp2
PP
pp1
pp2
Physics 151: Lecture 18, Pg 9
Elastic CollisionsElastic Collisions
Elastic means that energy is conserved as well as momentum.
This gives us more constraints.We can solve more complicated problems !!Billiards (2-D collision).The colliding objects
have separate motionsafter the collision as well as before.
Start with a simpler 1-D problem.
Before After
See text: 9.4
Physics 151: Lecture 18, Pg 10
Elastic Collision in 1-DElastic Collision in 1-D
v1,b v2,b
before
x
m1m2
See text: 9.4
v1,av2,a
afterm1
m2
Physics 151: Lecture 18, Pg 11
Elastic Collision in 1-DElastic Collision in 1-D
v1,b v2,b
v1,av2,a
before
after
x
m1 m2
Conserve PX
m1v1,b + m2v2,b = m1v1,a + m2v2,a
Conserve Energy
1/2 m1v21,b + 1/2 m2v2
2,b = 1/2 m1v21,a + 1/2 m2v2
2,a
Suppose we know v1,b and v2,b
We need to solve for v1,a and v2,a
Should be no problem 2 equations & 2 unknowns !
See text: 9.4
m1v1,b + m2v2,b = m1v1,a + m2v2,a
Physics 151: Lecture 18, Pg 12
Elastic Collision in 1-DElastic Collision in 1-D
After some moderately tedious algebra, (see text book Chapter 9, section3) we can derive the following equations for the final velocities,
See text: 9.4
BBAv
mmm
vmmmm
v2
21
2
1
21
21
1)
2()(
BBAv
mmmm
vmmm
v2
21
12
1
21
1
2)()
2(
1) m1v1,b + m2v2,b = m1v1,a + m2v2,a
2) v1,b - v2,b = - (v1,a - v2,a)
In general:
Physics 151: Lecture 18, Pg 13
Example - Elastic CollisionExample - Elastic Collision
Suppose I have 2 identical bumper cars. One is motionless and the other is approaching it with velocity v1. If they collide elastically, what is the final velocity of each car ?
Note that this means,
m1 = m2 = m
v2B = 0
See text: 9.4
Animation
Physics 151: Lecture 18, Pg 14
Lecture 18,Lecture 18, ACT 3ACT 3Elastic CollisionsElastic Collisions
I have a line of 3 bumper cars all touching. A fourth car smashes into the others from behind. Is it possible to satisfy both conservation of energy and momentum if 2 cars are moving after the collision?All masses are identical, elastic collision.
A) Yes B) No
Before
After? Animation
Physics 151: Lecture 18, Pg 15
Example of 2-D elastic collisions:Example of 2-D elastic collisions:BilliardsBilliards
If all we know is the initial velocity of the cue ball, we don’t have enough information to solve for the exact paths after the collision. But we can learn some useful things...
See text: Ex. 9.11
Physics 151: Lecture 18, Pg 16
BilliardsBilliards
Consider the case where one ball is initially at rest.
ppa
ppb
FF PPa
before afterthe final direction of the
red ball will depend on
where the balls hit.
vvcm
See Figure 12-14
See text: Ex. 9.11
Physics 151: Lecture 18, Pg 17
We know momentum is conserved: ppb = ppa + PPa
We also know that energy is conserved:
Comparing these two equations tells us that:
BilliardsBilliards
pb2 = (ppa + PPa )2 = pa
2 + Pa2 + 2 ppa PPa
m2
P
m2
p
m2
p 2a
2a
2 b
ppa PPa = 0
and must therefore be orthogonal!
Or … one momentum must be zero.
ppa
ppb
PPa
Ppp 2a
2a
2 b
See text: Ex. 9.11
Physics 151: Lecture 18, Pg 18
BilliardsBilliards
The final directions are separated by 90o.
ppa
ppb
FF PPa
before after
vvcm
See text: Ex. 9.11
Physics 151: Lecture 18, Pg 19
Lecture 18 –Lecture 18 – ACT 4ACT 4Pool SharkPool Shark
Can I sink the red ball without scratching ?Ignore spin and friction.
See text: Ex. 9.11
A) Yes B) No C) More info needed
Physics 151: Lecture 18, Pg 20
Lecture 18Lecture 18 – ACT 4 – ACT 4Pool SharkPool Shark
From above, after the collision the two balls move off at right angles.
Thus if the red ball goes toward a pocket, so does the cue ball
See text: Ex. 9.11
B) No
Physics 151: Lecture 18, Pg 21
Billiards.Billiards.
More generally, we can sink the red ball without sinking the white ball – fortunately.
See text: Ex. 9.11
Animation