A skydiver, with her gear, weighs 75 kg. She jumps from a helicopter at an altitude of 2 500 metres, waits 20 seconds and then opens her parachute. Assume that her initial velocity is zero, and also assume the following regarding air resistance: air resistance during free-fall is proportional to the square of her velocity, with drag coefficient k1 = 0:1; air resistance with an open chute is proportional to her velocity (i.e. a linear relationship), with drag coefficient k2 = 22. How long will it take her to reach the ground?

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To calculate how long it will take the skydiver to reach the ground, we need to break down the different stages of her fall and calculate the time taken for each stage.

1. Free-fall without parachute:
During the free-fall without the parachute, the air resistance experienced by the skydiver is proportional to the square of her velocity. We can use Newton's second law of motion to calculate the acceleration experienced by the skydiver:
F = ma
mg - kv^2 = ma (where m is the mass, g is the acceleration due to gravity, k is the drag coefficient, and v is the velocity)
Simplifying the equation, we get:
a = g - (k1/m) * v^2

Using kinematic equations, we can find the time taken for the free-fall phase. We know that the initial velocity is zero and the initial altitude is 2,500 meters. The final altitude during free-fall will be zero when she reaches the ground. Let's call the time taken for free-fall as t1.

Using the kinematic equation:
s = ut + (1/2)at^2

0 = 0*t1 + (1/2)(g - (k1/m) * v^2) * t1^2

Simplifying further:
t1^2 = (2 * (2,500) * m) / (g * k1)

2. Falling with an open chute:
Once the parachute is opened, the air resistance is proportional to the velocity. Again using Newton's law, we can calculate the acceleration experienced by the skydiver:
F = ma
mg - k2v = ma
a = g - (k2/m) * v

We can use the same kinematic equation as above to find the time taken for this phase. Let's call the time taken for this phase t2. Assuming that the skydiver reaches the ground during this phase, we can use the equation:

0 = 0*t2 + (1/2)(g - (k2/m) * v) * t2^2

Simplifying further:
t2^2 = (2 * (0) * m) / (g * k2)

3. Total time:
The total time taken for the skydiver to reach the ground will be the sum of t1 and t2:

Total time = t1 + t2

Now, let's substitute the given values and calculate the time:
- Mass (m) = 75 kg
- Acceleration due to gravity (g) = 9.8 m/s^2
- Drag coefficient for free-fall air resistance (k1) = 0.1
- Drag coefficient for open chute air resistance (k2) = 22

Using these values, we can calculate t1 and t2 and then find the total time.