Question

In the blizzard of ’88, a rancher was forced to drop hay bales from and airplane to feed her cattle. The plane flew horizontally at 160km/hr and dropped the bales from a height of 80m above the flat range.

(a) (5 points) Write the Hamiltonian for the motion. (Ignore air resistance)

(b) (5 points) What quantities are cyclic and what does it mean for its momentum.

(e) (5 points) Find the trajectory y(x) (not y(t))

Answer 1

Kinetic energy of object is

potential energy (taking ground as a reference)

Hamiltonian is

B)

x is not present in the Hamiltonian so it is cyclic.

As a result momentum corresponding x will be constant.

C)

Using the relations for hamiltonian

Using these equations we have to find a differential equation with x as independent variable and y as dependent.

eliminate t

from (1) and (3)

from 2 and 3

integrate this

substitute in 4

v_x is already given.