Homework 3
Classical Mechanics (Phys701)
Due: September 26, 2011
Problem 421
A particle is thrown up vertically with initial speed
v
0
, reaches a maximum height and falls back
to ground.
Show that the Corilois deﬂection when it again reaches the ground is opposite in
direction, and four times greater in magnitude, than the Coriolis deﬂection when it is dropped t
rest from the same maximum height.
Solution:
Consider a particle thrown up from the ground. Its vertical velocity is
v
z
=
v
0

gt
. Total
travel time is
T
= 2
v
0
/g
. Coriolis force creates a small deviation from the original trajectory and can be
taken into account as a small perturbation which does not alter the vertical motion.
Then the Coriolis
acceleration creates a horizontal velocity
v
y
(
t
) =
∫
t
0
2
ωv
z
(
s
)
ds
= 2
ω
(
v
0
t

gt
2
2
)
where
x
axis is chosen along the projection
ω
of the Earth’s angular velocity on the horizontal plane. The
horizontal displacement is then given by
y
1
=
∫
T
0
v
y
(
t
)
dt
= 2
ω
(
v
0
T
2
2

gT
3
6
)
=
4
ωv
3
0
3
g
2
Next, consider the particle falling from the height. Its vertical velocity is
v
z
=

gt
and the time of ﬂight
is
T/
2. We get
v
y
=

2
ω
∫
t
0
gs ds
=

2
ω
gt
2
2
y
2
=

ω
∫
T/
2
0
gt
2
dt
=

ω
g
3
(
T
2
)
3
=

ωv
3
0
3
g
2
=

y
1
4
Problem 422
A projectile is fired horizontally along Earth’s surface. Show that to a first approximation the
angular deviation from the direction of the fire resulting from the Coriolis effect varies linearly
with time at a rate
ω
cos
θ
where
ω
is the angular velocity of Earth’s rotation and
θ
is the colatitude, the direction of
deviation being to the right in the northern hemisphere.
Solution:
Let us point
x
axis along the fire direction. The
z
axis is up, and
y
axis points to the left
with respect to the fire direction. The Coriolis acceleration is

2[
⃗ω
×
⃗v
] = (0
,
2
ωv
sin
α,
0)
where
α
is the angle between
⃗v
and
⃗ω
. It is easy to see that
α
=
π/
2

θ
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 Bazaliy
 mechanics, Coriolis Effect, Force, Work, Rotation, vy, coriolis force

Click to edit the document details