Megan bailed out of her plane at an altitude of 8000 feet, fell freely for 15 seconds, and then opened her parachute. Assume that the drag coefficients are a=0.1 for a fee fall and a=1.6 with a parachute. When did she land(in seconds)?
To determine when Megan lands, we need to consider her motion during free fall and when her parachute is open. Let's break it down step by step:
1. Calculate the time it takes for Megan to reach the ground during free fall.
During free fall, the drag coefficient is a = 0.1. The equation that relates the time of free fall (t_ff), the acceleration due to gravity (g), and the drag coefficient (a) is as follows:
t_ff = sqrt((2 * h) / (g * a))
Where:
h = initial altitude = 8000 feet
g = acceleration due to gravity = 32.2 feet/second^2 (approximate value)
Substituting the given values into the equation:
t_ff = sqrt((2 * 8000) / (32.2 * 0.1))
t_ff = sqrt(4975)
t_ff ≈ 70.5 seconds
Therefore, it takes approximately 70.5 seconds for Megan to reach the ground during free fall.
2. Calculate the time Megan spends under the parachute.
After 15 seconds of free fall, Megan opens her parachute. From this point on, the drag coefficient changes to a = 1.6.
To determine the time spent under the parachute (t_parachute), we can use the following equation, which relates t_parachute, the initial vertical speed (v_i), the acceleration due to gravity (g), and the drag coefficient (a):
t_parachute = ln(v_i / g) / a
During free fall, the initial vertical speed is equal to the speed gained from falling freely for 15 seconds. The equation for the speed (v_ff) during free fall is:
v_ff = g * t_ff
Substituting the known values:
v_ff = 32.2 * 70.5
v_ff ≈ 2271 feet/second
Now, we can calculate the time Megan spends under her parachute:
t_parachute = ln(2271 / 32.2) / 1.6
t_parachute ≈ 6.5 seconds
Therefore, Megan spends approximately 6.5 seconds under her parachute.
3. Calculate the total time Megan takes to land.
To find the total time Megan takes to land, we add the time spent during free fall to the time spent under the parachute:
t_total = t_ff + t_parachute
t_total ≈ 70.5 + 6.5
t_total ≈ 77 seconds
Therefore, Megan lands approximately 77 seconds after bailing out of her plane.