Question

A hot-air balloon is drifting in level flight due east at 2.9 m/s due to a light wind. The pilot suddenly notices that the balloon must gain 26 m of altitude in order to clear the top of a hill 120 m to the east.

GIVENS:

How much time does the pilot have to make the altitude change without crashing into the hill? = 41s

What minimum, constant, upward acceleration is needed in order to clear the hill? = 3.0*10^-2m/s^2

What is the horizontal component of the balloon’s velocity at the instant that it clears the top of the hill? = 2.9m/s

Answer 1

Distance between the balloon and the hill horizontally = d = 120 m

Altitude change required to cleat the top of the hill = h = 26 m

The balloon is drifting in level flight due east that is it has no vertical velocity.

Horizontal velocity of balloon = V_x0 = 2.9 m/s

Vertical velocity of balloon = V_y0 = 0 m/s

No horizontal force will act on the balloon hecne the horizontal acceleration of the balloon is zero that is the horizontal velocity remains constant.

Vertical acceleration of the balloon = a_y

The balloon will reach the hill in time 't'

Horizontally,

d = V_x0t (Horizontal acceleration is zero)

120 = 2.9xt

t = 41.38 sec

For minimum constant upward acceleration the pilot will have to increase the altitude by 26m in 41.38 sec to clear the hill.

Vertically,

h = V_y0t + a_y(t²)/2

26 = 0(41.38) + a_y(41.38²)/2

a_y = 3.03 x 10^-2 m/s²

Horiztontal component of balloon's velocity at the instant it clears the top of the hill is 2.9 m/s as the horizontal velocity of the balloon remains constant as there is no acceleration in the horizontal direction.

Time the pilot has to make the altitude change without crashing into the hill = 41.38 sec.

Minimum constant upward acceleration needed to clear the hill = 3.03 x 10^-2 m/s².

Horizontal component of balloon's velocity at the instant it clears the top of the hill = 2.9 m/s.