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05/11/09
Physics 13  Fall 07  G.R. Goldstein
1
Falling ball seen from platform
distance along platform (meters)
height (meters)
Elapsed time is 1 sec
V=100 Km/hr
=27.8 m/sec
0.2 sec
0.6 sec
0.8 sec
0.4 sec
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Physics 13  Fall 07  G.R. Goldstein
2
acceleration
•
Newton’s 1st Law:
body remains in uniform motion
unless acted upon by (net) force
•
Nonuniform means
v changes
(magnitude
or
direction)
•
Changing v over time is
acceleration
a
=
dv
dt
=
d
2
x
dt
2
•
a
due to gravity = 9.8 m/s
2
= g
always downward
•
Vertical motion on earth
05/11/09
Physics 13  Fall 07  G.R. Goldstein
3
Galilean relativity
•
Ball seen from platform follows trajectory for initial velocity in x
direction u
x
= 100 Km/hr
= V of train (at time t=0)
•
No force in x direction so u
x
of ball unchanged and x increases
x
= u
x
t
•
Gravity in y direction, so downward u
y
increases and y falls from
say h = 5.0 m to 0 via y = h  gt
2
/2
y
= 0
for
1
2
gt
2
=
h
So
t
=
2
h
g
=
10.
m
9.8
m
/sec
2
=1.0 sec
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Physics 13  Fall 07  G.R. Goldstein
4
Galilean relativity (cont’d)
•
Acceleration same in either Frame of Reference 
•
So
F
=m
a
= m
a
′
Newton’s 2nd Law unchanged
•
There is no way to determine which Frame
of Reference is “at rest” by doing
experiments!
•
(Invariance of Laws of Physics)
05/11/09
Physics 13  Fall 07  G.R. Goldstein
5
•
Ball seen from platform follows trajectory for initial
velocity u
x
= 100 Km/hr = V and u
y
=0,u
z
=0
x
(
t
) =
x
(
t
) +
Vt
u
x
(
t
) =
dx
dt
=
d
dt
(
x
(
t
) +
Vt
) =
u
x
(
t
) +
V
so
u
x
=
u
x
+
V
addition of velocities
Also
a
x
(
t
) =
d
2
x
dt
2
=
d
dt
(
u
x
(
t
) +
V
) =
a
x
(
t
)
Hence
F
=m
a
= m
a
′
 Newton’s 2nd Law unchanged
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Physics 13  Fall 07  G.R. Goldstein
6
Lateral bouncing ball in 2 inertial frames
S
′
: frame on train seen from above
L
V
x
′
y
′
Position of ball on upward path
x
′
=0, y
′
=u
y
′
t
until y
′
=L
So for round trip
t
R
=2L/u
y
′
Let u
y
′
=c
′
the speed of ball
and note that u
y
’= u
y
in S
S: platform frame seen from above
L
x
y
d
Vt
R
/2
L
2
+ (
Vt
R
/2)
2
= d
2
= L
2
(1+V
2
/u
y
2
)
Now d/(t
R
/2) = c
ball speed seen
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This note was uploaded on 03/27/2008 for the course PHY 13 taught by Professor Garyr.goldstein during the Fall '04 term at Tufts.
 Fall '04
 GaryR.Goldstein
 Physics, Acceleration

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