Vectors again. Equations for Motion Along One Dimension.
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Transcript of Vectors again. Equations for Motion Along One Dimension.
Review Equations for Motion Along One Dimension
dt
dx
t
xv
t
xv
t
ave
0lim
dt
dv
t
va
t
va
t
ave
0lim
Review Motion Equations for Constant Acceleration
•1.
•2.
•3.
•4.
atvv 0
221
00 attvxx
20vv
vave
xavv 220
2
Review 3 Laws of Motion If in Equilibrium
If not in equilibrium Change in Motion is Due to Force
Force causes a change in acceleration
0F
maF
Law of conservation of energy
Power
efficiency
Review
NCspringgravspringgrav WUUKUUK 000
aveave Fvt
WP
in
out
P
Pe
If an 18 wheeler hits a car, what direction will the wreckage move?
What is the force between the 18 wheeler and the car?
Collisions
Newtons 2nd law
Linear momentum
SI
Newton defined it as quantity of motion
Momentum
vmp
sN
s
mkg
dt
When an object collides with another, the forces on the object will momentarily spike before returning back to zero.
Impulse
We now define impulse, J, as the change in momentum of a particle during a time interval
SI unit
Impulse
pJ
tFpJ
tFpJ
aveyyy
avexxx
sNs
mkg
A ball with a mass of 0.40 kg is thrown against a brick wall. It hits the wall moving horizontally to the left at 30 m/s and rebounds horizontally to the right at 20 m/s. (a) find the impulse of the net force on the ball during the collision with the wall. (b) If the ball is in contact with the wall for 0.010s, find the average horizontal force that the wall exerts on the ball during impact.
Example
If there are no external forces acting on the system
Conservation of Momentum
0
0
dt
pd
dt
pdFF
F
ABBonAAonB
Change in momentum over time is zero
The sum of momentums is constant
Conservation of Momentum
0)(
dt
ppd
dt
pd
dt
pd BAAB
constpp BA
If there are no external forces acting on a system, Total Momentum of a system conserved
Conservation of Momentum
constpp BA
BABA pppp
00
pp
0
A marksman holds a rifle of mass 3.00 kg loosely such that it’ll recoil freely. He fires a bullet of mass 5.00g horizontally with velocity relative to the ground of 300 m/s. What is the recoil velocity of the rifle?
Example - Recoil
Two battling robots are on a frictionless surface. Robot A with mass 20 kg moves at 2.0 m/s parallel to the x axis. It collides with robot B, which has a mass of 12 kg. After the collision, robot A is moving at 1.0 m/s in a direction that makes an angle α=30o. What is the final velocity of robot B?
Example – 2D example
Example
sm
Bx
Bx
B
AxAAxABx
BxBAxAAxA
BxBAxAAxA
BxAxBxAx
xx
v
v
m
vmvmv
vmvmvm
vmvmvm
pppp
pp
89.112
30cos)1)(20()2(20
0
0
0
00
0
Example
sm
By
By
B
AyABy
ByBAyA
ByBAyA
ByAyByAy
yy
v
v
m
vmv
vmvm
vmvm
pppp
pp
83.012
30sin)1)(20(
0
00
0
Elastic Collisions – Collisions where the kinetic energies are conserved. When the particles are in contact, the energy is momentarily converted to elastic potential energy.
Conservation of Momentum and Energy
Inelastic Collisions – collisions where total kinetic energy after the collision is less than before the collision.
Completely Inelastic Collisions- When the two particles stick together after a collision.
Collisions can be partly inellastic
Conservation of Momentum and Energy
Collisions where two objects will impact each other, but the objects stick together and move as one after the collision.
Completely Inelastic Collisions
Momentum is still conserved Find v in terms of v0
Completely Inelastic Collisions
vmvmvmvm
vvv
vmvmvmvm
pppp
pp
BABBAA
BA
BBAABBAA
BABA
00
00
00
0
Assume Particle B is initially at rest
Completely Inelastic Collisions
BA
AA
BAAA
BABBAA
mm
vmv
vmvmvm
vmvmvmvm
0
0
00
Kinetic Energy
Completely Inelastic Collisions
BA
A
BA
AA
BA
AABABA
AA
mm
m
K
K
mm
vmK
mm
vmmmvmmK
vmK
0
20
2
02
200
)(2
1
)(2
1)(
2
1
2
1
If B is at rest
At the intersection, a yellow subcompact car with mass travelling 950 kg east collides with a red pick up truck with mass 1900 kg travelling north. The two vehicles stick together and the wreckage travels 16.0 m/s 24o E of N. Calculate the speed of each of the vehicles. Assume frictionless.
Examples – Young and Freedman 8.37
Young and Freedman 8.37
B
yBAyB
A
xBAxA
y
x
yByAyBB
xBxAxAA
BABBAA
m
vmmv
m
vmmv
vv
vv
vmvmvm
vmvmvm
vmvmvmvm
)(
)(
24cos
24sin
0
0
0
0
00
Young and Freedman 8.37
sm
B
yBAyB
sm
A
xBAxA
y
x
m
vmmv
m
vmmv
vv
vv
9.21)(
5.19)(
6.1424cos1624cos
51.624sin1624sin
0
0
The ballistic pendulum is an apparatus to measure the speed of a fast moving projectile, such as a bullet. A bullet of mass 12g with velocity 380 m/s is fired into a large wooden block of mass 6.0 kg suspended by a chord of 70cm. (a) Find the height the block rises (b) the initial kinetic energy of the bullet (c) The kinetic energy of the bullet and block.
Problem – Ballistic Pendulum
Velocity after impact
Kinetic energy after impact
Problem – Ballistic Pendulum
BA
AA
BAAA
BABBAA
mm
vmv
vmvmvm
vmvmvmvm
0
0
00
J
mm
vmK
BA
AA 73.1)(2
1 20
Kinetic energy after impact
Converted to potential at highest point
Problem – Ballistic Pendulum
)(2
1 20
BA
AA
mm
vmK
gymm
mm
vm
gymmU
mm
vmK
BABA
AA
BA
BA
AA
)()(2
1
)(
)(2
1
20
20
Problem – Ballistic Pendulum
my
y
mm
vm
gy
gymmmm
vm
BA
AA
BABA
AA
0293.0
)012.6)(8.9(2
))380(012.0(
)(2
1
)()(2
1
2
2
2
20
20
JK
K
mvK
Bullet
Bullet
BBullet
866
)380(012.02
12
1
2
2
Momentum and Energy are conserved Find v in terms of v0
Elastic Collisions
BBAABBAA
BABA
vmvmvmvm
pppp
pp
00
00
0
2220
20
00
0
2
1
2
1
2
1
2
1BBAABBAA
BABA
vmvmvmvm
KKKK
KK
If particle B is at rest
Elastic Collisions – One Dimension
BBAAAA
BABA
vmvmvm
pppp
pp
0
00
0
2220
00
0
2
1
2
1
2
1BBAAAA
BABA
vmvmvm
KKKK
KK
If particle B is at rest
Elastic Collisions – One Dimension
BBAAA
BBAAAA
BBAAAA
vmvvm
vmvmvm
vmvmvm
)( 0
0
0
2220
2220
2220
)(
2
1
2
1
2
12
1
2
1
2
1
BBAAA
BBAAAA
BBAAAA
vmvvm
vmvmvm
vmvmvm
If particle B is at rest
Substitute back
Elastic Collisions – One Dimension
BAA
BBAAA
BBAAA
vvv
vmvvm
vmvvm
0
2220
0
)(
)(
)(
)(
)()(
)()(
0
0
00
00
00
BA
ABAA
ABAABA
ABAAABAA
ABABAAAA
AABAAA
mm
vmmv
vmmvmm
vmvmvmvm
vmvmvmvm
vvmvvm
If particle B is at rest
Elastic Collisions – One Dimension
)(
2
)(
)(
)(
)(
0
00
0
0
BA
AAB
BBA
ABAA
BAA
BA
ABAA
mm
vmv
vmm
vmmv
vvv
mm
vmmv
If ma <<< mb
really small
Elastic Collisions – One Dimension
)(
2
)(
)(
0
0
BA
AAB
BA
ABAA
mm
vmv
mm
vmmv
B
AA
BA
AAB
AB
ABA
m
vm
mm
vmv
vm
vmv
00
00
2
)(
2
If ma>>>mb
If ma=mb
Elastic Collisions – One Dimension
00
00
22
AA
AAB
AA
AAA
vm
vmv
vm
vmv
00
0
2
2
0)(
)0(
AA
AAB
BA
AA
vm
vmv
mm
vv
In a game of billiards a player wishes to sink a target ball in the cornet pocket. If the angle to the corner pocket is 35o, at what angle is the cue ball deflected? (Assume frictionless)
Example
Example
55
3590
0cos
cos20
20
2
)()(222
0
20
0020
2220
0
BA
BA
BBAAA
BABAA
AAA
BAA
BAA
vv
vv
vvvvv
vvvvv
vvv
vvv
vvv
Two particles with masses m and 3m are moving towards each other along the x axis with the same initial speeds. Particle m is travelling towards the left and particle 3m is travelling towards the right. They undergo an elastic glancing collision such that particle m is moving downward after the collision at right angles from initial direction. (a) Find the final speeds of the two particles. (b) What is the angle θ at which particle 3m is scattered.
Problem – Serway 9-36
Elastic Collisions and relative velocity – One Dimensional
ABBA
BBAA
BBBAAA
BBBBAAAA
BBBAAA
BBBBAAAA
BBAABBAA
BBAABBAA
vvvv
vvvv
vvmvvm
vmvmvmvm
vvmvvm
vmvmvmvm
vmvmvmvm
vmvmvmvm
00
00
20
2220
20
2220
00
00
2220
20
00
)()(
)()(
In an elastic Collision, the relative velocities of the two objects have the same magnitude
Elastic Collisions and relative velocity
ABAB
ABEAEB
EAEBEAEB
EAEBEBEA
ABBA
vv
vvv
vvvv
vvvv
vvvv
|0|0
|||
|||0|0
|||0|0
00
)(
A 0.150 kg glider (puck on an air hockey table) is moving to the right with a speed of 0.80 m/s. It has a head-on collision with a 0.300 kg glider that is moving to the left with velocity 2.20 m/s. Find the final velocities of the two gliders. Assume elastic collision.
Young and Freedman 8.42
A bat strikes a 0.145kg baseball. Just before impact the ball is travelling horizontally to the right at 50.0 m/s and it leaves the bat travelling to the left at an angle of 30o above the horizontal with a speed of 65.0 m/s. Find the horizontal and vertical components of the average force on the ball if the ball and bat were in contact for 1.75 ms.
Problems- Young and Freedman 8.12
A 23 g bullet travelling at 230 m/s penetrates a 2.0kg block of wood and emerges cleanly at 170 m/s. If the wood is initially stationary on a frictionless surface, how fast does it move after the bullet emerges?
Giancoli 7-12
A 90.0 kg full back running east with a speed 5.0 m/s is tackled by a 95.0kg opponent running north at 3.00 m/s. If the collision is completely inelastic, (a) find the velocity of the players just after the tackle. (b) find the mechanical energy lost during the collision.
Serway 9.28
Giancoli 7-78 A 0.25kg skeet (clay target) is fired at an
angle of 30o to the horizon with a speed of 25 m/s. When it reaches its maximum height, it is hit from below by a 15g pellet traveling vertically upwards at 200 m/s. The pellet is embedded into the skeet. (a) how much higher does the skeet go up? (b) how much further does the skeet travel?
Prepare for pain
Objects approximated to be point particles Objects only undergo translational motion
Assumptions so far
Real objects also undergo rotational motion while undergoing translational motion.
But there is one point which will move as if subjected to the same net force.
We can treat the object as if all its mass was concentrated on a single point
Center of Mass
Set an arbitrary origin point
Center of mass is the mass weighted average of the particles
Center of Mass
i
iicm m
mr
mmm
mrmrmrr
...
...
321
332211
A simplified water molecule is shown. The separation between the H and O atoms is d=9.57 x10-11m. Each hydrogen atom has a mass of 1.0 u and the oxygen atom has a mass of 16.0 u. Find the position of the center of mass.
Example
For ease set origin to one of the particles
Example
ohh
ohh
i
iicm
ohh
ohh
i
iicm
i
iicm
mmm
mdmdm
m
myy
mmm
mdmdm
m
mxx
m
mr
mmm
mrmrmrr
)0()5.52sin()5.52sin(
)0()5.52cos()5.52cos(
...
...
321
332211
Example
0)0()5.52sin()5.52sin(
105.6)5.52cos(2 12
ohh
ohh
i
iicm
ohh
h
i
iicm
mmm
mdmdm
m
myy
mxmmm
dm
m
mxx
1) if there is an axis of symmetry, the center of mass will lie along the axis.
2) the center of mass can be outside of the body
Center of Mass
The point of an object which gravity can be thought to act.
This is conceptually different from center of mass
For now the center of gravity of an object is also it’s center of mass.
Center of Gravity
Motion of Center of Mass
pmvmv
m
mv
mmm
mvmvmvv
dt
mrmrmrd
mmmdt
rd
mmm
mrmrmr
dt
d
dt
rd
m
mr
mmm
mrmrmrr
iiicm
i
iicm
cm
cm
i
iicm
...
...
...)(
...
1
...
...
...
...
321
332211
332211
321
321
332211
321
332211
External Forces and Center of Mass
cmiext
ernalexternal
iicmi
i
iicm
cm
cm
amF
FFF
Fmaam
m
ma
mmm
mamamaa
mmm
mvmvmv
dt
d
dt
vd
mmm
mvmvmvv
)(
)(
...
...
...
...
...
...
int
321
332211
321
332211
321
332211
James and Ramon are standing 20.0 m apart on a frozen pond. Ramon has a mass of 60.0 kg and James has mass of 90.0 kg. Midway between the two is a mug of their favourite beverage. They pull on the ends of a light rope. When James has moved 6.0 m how far has Ramon moved?
Example
Center of Mass will not move! James moved 6m to the right
mr
x
mx
m
mrr
cm
i
iicm
160
)90(4)6090(
26090
)60()90(4
A 1200 kg station wagon is moving along a straight highway at 12.0 m/s. Another car with mass 1800kg and speed 20.0 m/s has its center of mass 40.0 m away. (a) Find the position of the center of mass of the two cars. (b) Find magnitude of total momentum of the system. (c) Find the speed of the center of mass of the system. (d) Find total momentum using center of mass.
Problem – Young and Freedman 8.50