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10. Free fall
Free fall acceleration: g=9.8m/s2
Using general equations:
atvv
attvxx
0
2
00 2
Substitute:
yx
ga
To derive the following equations:
gtvv
gttvyy
0
2
00 2
2
2
0
20
20
vv
t
yv
vvyyg
2
2
0
20
20
vv
t
xv
vvxxa
1
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Example: A rocket is fired at a speed of 100 m/s straight up. (Neglect air resistance.)a)How long does it take to go up to the highest point?b)What is the maximum height?c)How long does it take to return back (go up and fall back down)?d)What is the speed of the rocket just before it falls down?
gvt
tgttv
gttvyy
c 0
2
0
2
00 2
0
20
2
0yy
0000 2 vvgvgvgtvv c
Solution:
0v
./1000 smv
a) At the highest point
;0 000 gvtgtvgtvv a ssmsmta 2.108.9/100 2
c) When it comes back
;
222
20
20
0
2
00max g
vgvgtv
gttvyyh a
aa
Given:
m
sm
smh 510
8.92
/1002
2
max
b) The maximum height is
d) The speed it falls down with is
stt ac 4.202
2
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The trajectory of an object projected with an initial
velocity at the angle above the horizontal
with negligible air resistance.0v
0
11. Projectile Motion
sin
cos
00
00
vv
vv
y
x
smga
constvv xx
/8.90
3
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Shoot the monkey (tranquilizer gun)
A zookeeper shoots a tranquilizer dart to a monkey that hangs from a tree. He aims at the monkey and shoots a dart with an initial speed v0.
The monkey, startled by the gun, lets go immediately. Will the dart hit the monkey?
A. Only if v0 is large enough.
B. Yes, regardless of the magnitude of vo.
C. No, it misses the monkey.
If there is no gravity, the dart hits the monkey…
If there is gravity, the dart also hits the monkey!
4
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If there is no gravity, the dart hits the monkey…
Continued
5
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If there is gravity, the dart also hits the monkey!
Note, that it takes the same amount of time to hit the monkey as in the no gravity case!
Continued
6
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This might be easier to think about…
2
0
2
1gty
tvx
For the bullet:
2
0
2
1gty
xx
For the monkey:
Continued
7
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Example: A rocket is fired at a speed of 100 m/s at an angle 30° above the horizontal . (Neglect air resistance.)a)What are the initial values of the x and y components of the speed?b)How long is the rocket be in the air?c)What is the distance between the lunch and landing points (assuming that these points have the same altitude)?
Solution:
smv /1000 Given:
a) smsmvv
smsmvv
y
x
/0.5030sin/100sin
/6.8630cos/100cos
00
00
b) From the example on slide 2, question (c) follows that
;2 0 gvt y ssmsmt 2.108.9/502 2
c) The horizontal motion is uniform, and the distance is
mssmtvd x 8832.10/6.860
0v
xv0
yv0
x
y
8
30
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12. Circular motion v
r
d
x
y
r
r
varad
2
frequencyangular - 2
frequency - 1
πf
Tf
Definitions:
For any circular motion there is radial (centripetal) acceleration:It is directed along radius and to the center.
For uniform circular motion (v=const):
rfrT
r
t
dv
22
T – period (time it takes to return to the same point)
rarad2
9
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t
va
t
0
tan lim
22tan radaaa
Optional information (if you are curious):
If circular motion is not uniform (v ≠ const) then there is tangential acceleration:
The tangential acceleration is perpendicular to the radial acceleration, and the total acceleration is equal to:
12a. Circular motion (continues)
10
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Example: Period of a satellite motion
g
R
kmR
smg
6400
/8.9 2
ga
R
va
2
gR
v
2
gRv g
R
v
RT
22
min83
min60
500050008002
/8.9
/100064002
2
sss
sm
kmmkmT
skmsmkmmkmsmgRv /8/108/10006400/8.9 32
11
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12
2,1
2,1
2
1
?
?
20
40.0
20.0
rada
v
T
rpmf
mr
mr
sf
TTss
f 31
3
1
60
min1
min
120 21
sms
mv
sms
mv
/84.03
40.02
/42.03
20.02
2
1
T
r
t
dv
2
Example: Two balls attached to a string as shown at 0.20 m and 0.40 m from the center move in circles at a uniform frequency of 20 rpm. What are their periods, linear speeds, and radial accelerations?
rfarad22
22
2
22
1
/76.140.03
2
/88.020.03
2
smms
a
smms
a
rad
rad
12