To determine the acceleration of the airplane 25s after the start of the climb, we need to break it down into two components: tangential and radial.
1. Tangential acceleration (at):
The tangential acceleration can be found using the formula:
at = v' * t,
where v' is the rate of increase of speed and t is the time.
Given:
v' = 20 ft/s^2,
t = 25s.
Substituting the values into the formula, we get:
at = 20 ft/s^2 * 25s = 500 ft/s.
2. Radial acceleration (ar):
The radial acceleration can be found using the formula:
ar = v^2 / p,
where v is the instantaneous velocity and p is the radius of curvature.
Given:
v = vo + at * t,
vo = 150 mph (convert to ft/s),
at = 500 ft/s,
t = 25s,
p = 2000 ft.
Converting the initial velocity to ft/s:
vo = 150 mph * 5280 ft/mile * (1/3600) hour/s = 220 ft/s.
Substituting the values into the formula, we get:
v = 220 ft/s + 500 ft/s * 25s = 12720 ft/s.
Substituting the values into the formula for radial acceleration, we get:
ar = (12720 ft/s)^2 / 2000 ft = 81264 ft/s^2.
Therefore, the acceleration of the airplane 25s after the start of the climb is:
ar = 81264 ft/s^2 (radial component).
at = 500 ft/s (tangential component).