collisions f - facweb1.redlands.edu
TRANSCRIPT
Fri. 10.6-.8 Scattering RE 10.b
Mon.,
Lab
Wed.,
Fri.,
10.9-.10 Collision Complications
L10 Collisions 1
10.5, .11 Different Reference Frames
11.1 Translational Angular Momentum Quiz 10
RE 10.c
EP9
RE 11.a; HW10: Pr’s 13*, 21, 30, “39”
Collisions Short, Sharp Shocks
Smack!
BF
Which of the following is a property of all collisions - both “elastic” and
“inelastic” collisions?
(1) The internal energy of the system after the collision is different from what it
was before the collision.
(2) The total momentum of the system doesn’t change.
(3) The total kinetic energy of the system doesn’t change.
Which of the following is a property of all “elastic” collisions?
(1) The colliding objects interact through springs.
(2) The kinetic energy of one of the objects doesn’t change.
(3) The total kinetic energy is constant at all times -- before, during, and after the
collision.
(4) The total kinetic energy after the collision is equal to the total kinetic energy
before the collision.
(5) The elastic spring energy after the collision is greater than the elastic spring
energy before the collision.
1-D Collision
A B
Pow! A B
Example
System = carts A & B
initially
finally
frictionAF
frictionBF
frictionAF
frictionAF
Often know initial motion, want to predict final motion
Two unknowns: and fAv .
fBv .
Generally need two equations to solve for them
0& BABA ppp
True for all collisions
0&& BABABA UEEE
BB EK .intAA EK .int
A B
iAp .
iBp .
Pow! A B
fAp .
fBp .
System = carts A & B
initially
finally
0 BA pp
0.... iBBfBBiAAfAA vmvmvmvm
csv '
1-D Collision Special Case: “Maximally Inelastic” – hit & stick
0.. iBBfBiAAfA vmvmvmvmfBfAf vvv
BA
iBBiAAf
mm
vmvmv
..
A B
iAp .
iBp .
Pow! A B
fAp .
fBp .
System = carts A & B
initially
finally
02
.212
.212
.212
.21 iBBfBBiAAfAA vmvmvmvm csv '
1-D Collision
BABBAABA UEKEKE &int.int.&
Special Case: Perfectly Elastic (all internal changes ‘bounce back’)
02222
2
.
2
.2
.
2
.
B
iB
B
fB
A
iA
A
fA
m
p
m
p
m
p
m
p
iBiAfBfA pppp ....
Equation 1
Equation 2
Extra special case: Say B is initially stationary
1-D Collision Special Case: Perfectly Elastic (all internal changes ‘bounce back’)
A B
Pow! A B
fAp .
fBp .
initially
finally
iA
AB
A vmm
m.
2
iA
AB
BA vmm
mm.
iAv .2
iAv .
1 iAv .
0
𝑣 𝐵.𝑓 𝑣 𝐴.𝑓
𝑚𝐵
𝑚𝐴
Initially Stationary
𝑣 𝐴.𝑓 𝑣 𝐵.𝑓
Transform for B initially moving
1-D Collision Special Case: Perfectly Elastic
A B
A B
Initially I see
Finally I see
𝑣 𝐴.𝑓.𝑚𝑒 𝑣 𝐵.𝑓.𝑚𝑒
Say I’m watching this collision while I’m just sitting here. According to me
meiAv ..
0.. meiBv
Say you’re watching this collision while walking by with velocity .
Relative to you the initial velocities are youv
A B Initially you see
youv
youmeiAyouiA vvv
.... youyouiB vv
..
Similarly, after the collision, according to me
And according to you
A B Finally I see
𝑣 𝐴.𝑓.𝑦𝑜𝑢 = 𝑣 𝐴.𝑓.𝑚𝑒 − 𝑣 𝑦𝑜𝑢 𝑣 𝐵.𝑓.𝑦𝑜𝑢 = 𝑣 𝐵.𝑓.𝑚𝑒 − 𝑣 𝑦𝑜𝑢
youv
Transform for B initially moving
1-D Collision Special Case: Perfectly Elastic
So, completely rephrasing things from your perspective:
A B Initially you see
youv
youmeiAyouiA vvv
.... youyouiB vv
..
A B Finally I see
𝑣 𝐴.𝑓.𝑦𝑜𝑢 = 𝑣 𝐴.𝑓.𝑚𝑒 − 𝑣 𝑦𝑜𝑢 𝑣 𝐵.𝑓.𝑦𝑜𝑢 = 𝑣 𝐵.𝑓.𝑚𝑒 − 𝑣 𝑦𝑜𝑢
youv
youiBmeiAyouiA vvv ......
youiByouiAmeiA vvv ......
youiBmefAyoufA vvv ......
youiBmefAyoufB vvv ......
youiBmeiA
AB
BAyoufA vv
mm
mmv ......
iBiBiA
AB
BAfA vvv
mm
mmv ....
youiBmeiA
AB
AyoufB vv
mm
mv ......
2
iBiBiA
AB
AyoufB vvv
mm
mv .....
2
P
mp
ipp .
T
mT
0. iTp
Slow (v<<C)
2-D Collision: Scattering
P
mp
0,0,.. ipip pp
T
mT
0. iTp
T
P qp
qT
00sinsin:ˆ
0coscos:ˆ
0
..
...
...
TfTpfp
ipTfTpfp
ipfTfp
ppy
pppx
ppp
Momentum Principle Energy Principle
x
y
0,sin,cos ... pfppfpfp ppp qq
0,sin,cos ... TfTTfTfT ppp qq
Slow (v<<C)
2-D Collision: Scattering
Three Equations / Four Unknowns
Need another equation
0
0
,.int.int...
....
pttpipfTfp
iTipfTfp
UEEKKK
EEEE
0222
2
.
2
.
2
.
p
ip
T
fT
p
fp
m
p
m
p
m
p if Elastic
Or more information
P
mp
0,0,.. ipip pp
T
mT
0. iTp
T
P qp
qT
00sinsin:ˆ
0coscos:ˆ
..
...
TfTpfp
ipTfTpfp
ppy
pppx
qqMomentum Principle
Energy Principle – if Elastic
0222
2
.
2
.
2
.
p
ip
T
fT
p
fp
m
p
m
p
m
p
x
y
0,sin,cos ... pfppfpfp ppp qq
0,sin,cos ... TfTTfTfT ppp qq
2-D Collision: Scattering Example. Say a 2kg puck moving at 3m/s collides with a 4kg puck that’s just sitting
there. Say the 2kg puck travels off at 30° and the 4kg one travels down at 50°. What
are the final speeds and is the collision elastic?
P
mp
0,0,.. ipip pp
T
mT
0. iTp
T
P qp
qT
fTfpip ppp ...
T
fT
p
fp
p
ip
m
p
m
p
m
p
222
2
.
2
.
2
.
x
y
0,sin,cos ... pfppfpfp ppp qq
0,sin,cos ... TfTTfTfT ppp qq
Slow (v<<C), Elastic Collision
2-D Collision: Scattering
Three Equations / Four Unknowns
Need another equation
Special case: equal masses
2
.
2
.
2
. fTfpip ppp
Tp mm
tfpftfpfpipipi ppppppp
2
TfpfTfpfpi ppppp
2222
compare
Must be zero
A: projectile missed target
TpTfpfTfpfpi ppppp qq cos2222
B: qp + |qT| = 90°
P
mp
0,0,.. ipip pp
T
mT
0. iTp
T
P qp
qT
00sinsin:ˆ
0coscos:ˆ
0
..
...
...
TfTpfp
ipTfTpfp
ipfTfp
ppy
pppx
ppp
Momentum Principle Energy Principle – if Elastic
0222
0
0
2
.
2
.
2
.
...
....
p
ip
T
fT
p
fp
ipfTfp
iTipfTfp
m
p
m
p
m
p
KKK
EEEE
x
y
0,sin,cos ... pfppfpfp ppp qq
0,sin,cos ... TfTTfTfT ppp qq
Slow (v<<C), Elastic Collision
2-D Collision: Scattering
Three Equations / Four Unknowns
Need another equation
P mp
0,0,.. ipip pp
T mT
0. iTp
T
P qp
qT
x
y
0,sin,cos ... pfppfpfp ppp qq
0,sin,cos ... TfTTfTfT ppp qq
Slow (v<<C), Elastic Collision
2-D Collision: Scattering
Collision geometry and Impact Parameter
Three Equations / Four Unknowns
Need another equation
P
dt
Component of momentum along line of
contact changes, other component
remains constant
b
b = Impact Parameter
00sinsin:ˆ
0coscos:ˆ
0
..
...
...
TfTpfp
ipTfTpfp
ipfTfp
ppy
pppx
ppp
Momentum Principle Energy Principle – if Elastic
0222
0
0
2
.
2
.
2
.
...
....
p
ip
T
fT
p
fp
ipfTfp
iTipfTfp
m
p
m
p
m
p
KKK
EEEE
Which of these is true?
1) The larger the impact parameter, the larger the scattering angle (deflection).
2) The larger the impact parameter, the smaller the scattering angle (deflection).
P
T T b
P mp
T mT
T
P
2-D Collision: Scattering
P
b
Qualitative effect of Impact Parameter
Miss: b too big
Glancing Blow: b big P mp
T mT
T
P
b
P
Solid Blow: b small
No deflection
minor deflection
P mp
T mT
P b
P major deflection
T
Probability and Cross-sectional Area
Playing Darts Badly
2-D Collision: Scattering – Discovering
Nucleus
In the Rutherford experiment, what was surprising to the experimenters?
(1) Sometimes the alpha “rays” passed right through the gold foil.
(2) Sometimes the alpha “rays” were deflected slightly when they passed through
the gold foil.
(3) Sometimes the alpha “rays” bounced back from the gold foil.
• relativistic speeds
• identifying ‘mystery’ particles
• the mathematical trick of analyzing in
the center-of-mass reference frame
Deeper Collisions
http://en.wikipedia.org/wiki/File:CDF_Top_Event.jpg
P
mp
0,0,.. ipip pp
T
mT
0. iTp
T
P
0,sin,cos ... pfppfpfp ppp qq
qp
0,sin,cos ... TfTTfTfT ppp qq
qT
Conservation of Momentum
Conservation of Energy
0.... iTipfTfp EEEE
where 2
1cv
vmp
00sinsin:ˆ
0coscos:ˆ
0
..
...
...
TfTpfp
ipTfTpfp
ipfTfp
ppy
pppx
ppp
where 2
2
1cv
mcE
show
By plugging that into it and
recovering that.
222mcpcE
2-D Collision: Scattering
Fast
02
.
22
.
2
.
22
.
2
.
22
.
2
. cmcmcpcmcpcmcp iTipipfTfTfpfp
Note: initial and final masses differ for inelastic collisions
2-D Collision: Scattering
Fast Note: initial and final masses differ for inelastic collisions
int.ppp EKE
Recall: particle energy
2
int. cmE pp
Elastic
02
int. cmE pp
In-Elastic
02222
.
222
.
222
. cmcmcpcmcpcmcp TpipTfTpfp
02
int. cmE pp
02
.
22
.
2
.
22
.
2
.
22
.
2
. cmcmcpcmcpcmcp iTipipfTfTfpfp
2-D Collision: Fast Example: A beam of high energy p- (negative pions) is shot at a flask of liquid
hydrogen, and sometimes a pion interacts through the strong interaction with a proton
in the hydrogen, the reaction is where X+ is a positively charged
particle of unknown mass (essentially an excited proton.)
A proton’s rest mass is 938MeV, and a pion’s rest mass is 140 MeV. The incoming
pion has momentum 3GeV/c. It scatters through 40°, and its momentum drops to
1.510 GeV/c.
What is the rest mass of the X+?
02
.
222
.
22222
.
2
. cmcmcpcmcpcmcp ipiXXff pppp
p p p X
0sinsin:ˆ
0coscos:ˆ
..
...
XfXf
iXfXf
ppy
pppx
pp
ppp
p
mp
ip .p
p
mp
0. ipp
X
p
fp .p
qp
fXp .
qX Conservation of Momentum
Conservation of Energy
pp
ppp
sinsin
coscos
..
...
fXfX
fiXfX
pp
ppp
2
.
2
..
2
.
2
. sincossincos ppppp qqqq ffiXfXXfX ppppp
2
.
2
... sincos ppppp qq ffifX pppp ppppp qcos2 ..
2
.
2
.. fififX ppppp
2-D Collision: Fast Example: A beam of high energy p- (negative pions) is shot at a flask of liquid
hydrogen, and sometimes a pion interacts through the strong interaction with a proton
in the hydrogen, the reaction is where X+ is a positively charged
particle of unknown mass (essentially an excited proton.)
A proton’s rest mass is 938MeV, and a pion’s rest mass is 140 MeV. The incoming
pion has momentum 3GeV/c. It scatters through 40°, and its momentum drops to
1.510 GeV/c.
What is the rest mass of the X+?
02
.
222
.
22222
.
2
. cmcmcpcmcpcmcp ipiXXff pppp
p p p X
p
mp
ip .p
p
mp
0. ipp
X
p
fp .p
qp
fXp .
qX
Conservation of Momentum
Conservation of Energy
ppppp qcos2 ..
2
.
2
.. fififX ppppp
22
.
2
.
2
.
222
.
222cmcpcmcmcpcmcp ffipiXX pppp
Could do algebraically further, or just plugin numbers and simplify
Fri. 10.6-.8 Scattering RE 10.b
Mon.,
Lab
Wed.,
Fri.,
10.9-.10 Collision Complications
L10 Collisions 1
10.5, .11 Different Reference Frames
11.1 Translational Angular Momentum Quiz 10
RE 10.c
EP9
RE 11.a; HW10: Pr’s 13*, 21, 30, “39”
Collisions Short, Sharp Shocks
Smack!
BF
