Chapter 10 Collisions. 2 7/19/04 Review Momentum: If F ext = 0, then momentum does not change For...
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Transcript of Chapter 10 Collisions. 2 7/19/04 Review Momentum: If F ext = 0, then momentum does not change For...
Chapter 10
Collisions
27/19/04
Review
Momentum:
vmp
If Fext = 0, then momentum does not change
fi pp
f
iexif M
M+v=vv ln
For continuous momentum transfer (Rockets):
37/19/04
Rockets: Continuous Momentum Transfer
47/19/04
Momentum in a Collision
Total momentum is conserved
In a collision, objects only exert forces on each other, so Fext=0.
57/19/04
Impulse
During a collision, the momentum on an object changes
This change in momentum is called “Impulse”
When objects A and B collide
if pppJ
BA JJ
67/19/04
Impulse
Recall:
t
pF
tFpJ
In the limit of small t:
f
i
f
i
t
t
p
pdtFpdJ
(constant force)
(changing force)
77/19/04
Impulse in a Collision
Different collisions with the same total impulse:
Blue
Red
Large F: p changes rapidly
Small F: p changes slowly
p/t Large
p/t Small Mo
men
tum
t
pF
87/19/04
Example: The Impulsive Spiderman
Spiderman, who has a mass of 70 kg, jumps from a train 5 meters high moving at 20 m/s (about 40 mph).
He lands standing up, taking t = 0.1 s to stop himself after making contact with the ground. How much force did his knees feel?
97/19/04
ExampleTreat as collision between Spiderman and the ground
Initial: p = mvtotal
Final: p = 0
Get force from the impulse:
t
mv
t
pF total
tFpJ
107/19/04
Example
t
mv
t
pF total
ghv
mghmv
KU
y
y
finalinitial
22
2
21
Need to find vy:
sm
msmsm
vvv yxtotal
/3.22
)5)(/8.9(2)/20( 22
22
Ns
smkgF 000,16
1.0
)/3.22)(70(
If he wasn’t a superhero, he’d break his legs!
117/19/04
What if he rolls on landing for t = 2 sec?
Much easier on the knees!
Example
Ns
smkgF 780
2
)/3.22)(70(
127/19/04
Cannon RecoilCannon: mc=1134 kg
Ball: mb=13.6 kg
Ball shot at ~ speed of sound vb = 340 m/s
The cannon and ball are initially at rest:
ballcannon
cannonball
fi
pp
pp
pp
0
0
pball = mballvball = (13.6kg)(340 m/s) = 4620 kg m/s
So, pcannon= -4620 kg m/s
137/19/04
Cannon Recoil
Cannon recoil stopped in ~2 s by ropes. What is the tension in the ropes?
A rope can easily handle this much force withoutbreaking
TT
pc
t
pTFnet
2
lbsNs
smkg
t
pT 260~1160
2
/4620
2
1
2
1
147/19/04
Momentum Conservation in Different Frames
m mv -v
Simple 1D problem
PTOT = mv - mv =0
2m
Sticktogether v=0
157/19/04
Momentum Conservation in Different Frames
m m2v
Same 1D problem viewed from right hand block, or with right hand block at rest
PTOT = 2mv + 0 = 2mv
2 m v
167/19/04
Changes in Momentum Independent of Frame
Left
Right
Case 1 Case 2
i f i f
mv
-mv
0
0
2mv
0
mv
mv
PTf – PTi = 0 – 0 = 0 PTf – PTi = 2mv – 2mv = 0
177/19/04
Center of Momentum Frame
There is always a frame of reference where PTOT=0.
‘Center of mass’ frame
187/19/04
A Limitation of Momentum
V=30 MPH
Before
V=0
m/s kg,
m/s. kg,
MPH lbsp
= m/s kg
MPH) lbsp
truck,i
car,i
29221
)413)(5891(
)30)(3500(
0)0)(681(
)0)(1500(
m/s kg,
ppp truck,icar,itotal,i
29221
truck,fcar,ftotal,f p pp
How do we determine
the velocities?
After
vT vc
BOOM!
197/19/04
A Limitation of MomentumConstant truck,fcar,ftotal,f ppp
truck,ftotal,f car,f ppp
There are many possibilities
Conservation of Momentum can’t tell them apart
ptruck
p ca
r
207/19/04
Elastic Collisions
Momentum and kinetic energy are conserved
Two equations:
Good approximation for a lot of collisions, and exact for some
Examples: Billiard Balls, superball on floor…
2,222
12,112
12,222
12,112
1
,22,11,22,11
ffii
ffii
vmvmvmvm
vmvmvmvm
217/19/04
Elastic Collisions in One Dimension
Two conservation laws
Momentum
Energy
(Elastic only - Mechanical Energy is conserved)
m1 m2
Before
V1,i V2,i
m1 m2
After
V1,f V2,f
(Always)
2,222
12,112
12,222
12,112
1ffii vmvmvmvm
ffii vmvmvmvm ,22,11,22,11
227/19/04
A Unique Solution
,i,i,f
,i,i,f
vmm
mmv
mm
mv
vmm
mv
mm
mmv
221
121
21
12
221
21
21
211
2
2
We now have two equations and two unknowns:
Lots of Algebra
2,222
12,112
12,222
12,112
1ffii vmvmvmvm
ffii vmvmvmvm ,22,11,22,11
237/19/04
Limiting Cases
vv
vv
,i,f
,i,f
12
21
m1 = m2
How do we understand what types of motion
these predict?
Consider limiting case:
The two objects simply trade values of velocity!
,i,i,f
,i,i,f
vmm
m mv
mm
mv
vmm
mv
mm
mmv
221
121
21
12
221
21
21
211
2
2
247/19/04
Limiting Cases
What if m1 >> m2?
,i,i,f
,i,f
vvv
vv
212
11
2
Semi truck hits a parked VW bug: Truck keeps going
Bug bounces off with twice truck’s speed!
,i,i,f
,i,i,f
vmm
m mv
mm
mv
vmm
mv
mm
mmv
221
121
21
12
221
21
21
211
2
2
7/19/04
Demonstration
m1>>m2
0
0
vv
vv
tennis,i
basket,i
A Question:
WhatHappens?Before:
After:
0
0
3vv
vv
tennis,f
basket,f
,i,i,f
,i,f
vvv
vv
212
11
2
26
The Slingshot Effect
km/s.
km/s km/s.
vvv probe,ijupiterprobe,f
229
)10()69(2
2
-10 km/s
9.6 km/s
277/19/04
Car-Truck Crash
A 2000 kg car has a head-on collision with a 10,000 kg truck. They each are travelling at 10 m/s and they collide elastically (solid bumpers!).
m1 m2v1i v2i
What are their final velocities?
Choose positive x direction +x
287/19/04
Car-Truck Crash (continued)
m1 m2v1i v2i
v1i = 10 m/s
v2i = -10 m/s
m1 = 2,000 kg
m2 = 10,000 kg
v1f = -23.3 m/sv2f = -3.33 m/s
Truck slows down Car goes flying backwards!
,i,i,f
,i,i,f
vmm
mmv
mm
mv
vmm
mv
mm
mmv
221
121
21
12
221
21
21
211
2
2
297/19/04
Car-Truck Crash (continued)
If the two vehicles are being driven by 60 kg PSU students, what are the impulses they feel?
In truck: J = p = mv = m(v2f - v2i)
= 60(-3.33 – (-10))
= 400 kg m/s
= m(v1f – v1i)
= 60(-23.3 – (10))
= -2000 kg m/s
In car: J = p = mv
307/19/04
Car-Truck (question)
Which would you rather be driving?
Say collision lasts Δt = 0.2 seconds
Force on student is given by F = Δp/Δt
Student in truck feels 2,000 N (survivable)
Student in car feels 10,000 N (not good)
What if instead of a 2000 kg car, she was on a 500 kg motorcycle!
317/19/04
Example: 2-D Elastic Collision
Two billiard balls collide elastically on a table. The initial velocity of the first ball is v1,i=(1 m/s)i+(2 m/s)j. The second ball is initially at rest. Both balls have the same mass. Determine the final velocity of both after the collision.
v1,i=(1 m/s)i+(2 m/s)j
327/19/04
Inelastic Collisions
Momentum is conserved (NOT Kinetic Energy)
Examples: Spit wads, football player being tackled,…
fii vmmvmvm
)( 21,22,11
Two objects stick together
Completely Inelastic:
337/19/04
Inelastic Collisions…
http://www.baylortv.com/streaming/000026/300kbps_ref.mov
347/19/04
Car Crash
v1=20 m/s v2=30 m/s
m1=750 kg m2=1000 kg
Two cars collide and stick together after the collision. What is the final velocity of the system?
357/19/04
Car Crash
v1=20 m/s v2=30 m/s
m1=750 kg m2=1000 kg
Using conservation of momentum:
smv
kgkg
smkgsmkg
mm
vmvmv
vmmvmvm
pp
f
iif
fii
fi
/6.8
)1000()750(
)/30)(1000()/20)(750(
)(
21
,22,11
21,22,11
367/19/04
Basketball Cannon
A ball projected from a cannon hits the trash can such that:
1) It sticks into the trash can.2) It hits the trash can and bounces back.
Will the velocity of the trash can be bigger for case 1, case 2, or exactly the same?
377/19/04
Basketball Cannon
v
m
vtrash=0
M
Consider an elastic collision:
vmM
mv
mM
mMv trash,itrash,f
2
mM
mvvtrash,f
2
387/19/04
Basketball Cannon
v
m
vtrash=0
M
Consider a perfectly inelastic collision:
fvmMmv )(
mM
mvv f
397/19/04
Basketball Cannon
Elastic:
mM
mvv f
2
Inelastic:
mM
mvv f
Elastic collision results in twice the velocity!