Introduction - An Introduction to Dynamic Meteorology - page 25

1.8. Find the horizontal displacement of a body dropped from a fixed platform
at a height h at the equator neglecting the effects of air resistance. What is
the numerical value of the displacement for h = 5 km?

1.9. A bullet is fired directly upward with initial speed w 0 , at latitude φ. Neglect-
ing air resistance, by what distance will it be displaced horizontally when
it returns to the ground? (Neglect 2u cosφ compared to g in the vertical
momentum equation.)

1.10. A block of mass M 1 kg is suspended from the end of a weightless string.

The other end of the string is passed through a small hole in a horizontal
platform and a ball of mass m

=

10 kg is attached. At what angular velocity
must the ball rotate on the horizontal platform to balance the weight of the
block if the horizontal distance of the ball from the hole is 1 m? While the
ball is rotating, the block is pulled down 10 cm. What is the new angular
velocity of the ball? How much work is done in pulling down the block?
1.11. A particle is free to slide on a horizontal frictionless plane located at a latitude
φ on the earth. Find the equation governing the path of the particle if it is
given an impulsive northward velocity v

=

0. Give the solution for
the position of the particle as a function of time. (Assume that the latitudinal
excursion is sufficiently small that f is constant.)

= V 0 at t =

1.12. Calculate the 1000- to 500-hPa thickness for isothermal conditions with
temperatures of 273- and 250 K, respectively.

1.13. Isolines of 1000- to 500-hPa thickness are drawn on a weather map using a
contour interval of 60 m. What is the corresponding layer mean temperature
interval?

1.14. Show that a homogeneous atmosphere (density independent of height) has
a finite height that depends only on the temperature at the lower boundary.
Compute the height of a homogeneous atmosphere with surface temperature
T 0

273K and surface pressure 1000 hPa. (Use the ideal gas law and
hydrostatic balance.)
1.15. For the conditions of Problem 1.14, compute the variation of the temperature
with respect to height.

=

1.16. Show that in an atmosphere with uniform lapse rate γ (where γ ≡− dT/dz)

the geopotential height at pressure level p 1 is given by

1

− Rγ /g p 0

p 1

T 0
γ

= −

Z

where T 0 and p 0 are the sea level temperature and pressure, respectively.