Skydiver/Physics Diff.Eq. Problem

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  • Thread starter alane1994
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In summary, the skydiver falls vertically downward at a speed of 0 ft/s and open the parachute after 10 s of free fall. The parachute has a weight of mg and produces a friction force of 0.75v which points upward. The skydiver falls a distance of 10 ft before the parachute opens.
  • #1
alane1994
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A skydiver weighing 180lb (including equipment) falls vertically downward from an altitude of 5000 ft and opens the parachute after 10 s of free fall. Assume that the force of air resistance, which is directed opposite to the velocity, is of magnitude \(0.75|v|\) when the parachute is closed and is of magnitude of \(12|v|\) when the parachute is open, where the velocity \(v\) is measured in ft/s.

a) Find the speed of the skydiver when the parachute opens.
b) Find the distance fallen before the parachute opens.
c) What is the limiting velocity \(v_L\) after the parachute opens?

Any and all assistance would be appreciated greatly. And thanks for putting up with all of my questions :eek:
 
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  • #2
alane1994 said:
A skydiver weighing 180lb (including equipment) falls vertically downward from an altitude of 5000 ft and opens the parachute after 10 s of free fall. Assume that the force of air resistance, which is directed opposite to the velocity, is of magnitude \(0.75|v|\) when the parachute is closed and is of magnitude of \(12|v|\) when the parachute is open, where the velocity \(v\) is measured in ft/s.

a) Find the speed of the skydiver when the parachute opens.
b) Find the distance fallen before the parachute opens.
c) What is the limiting velocity \(v_L\) after the parachute opens?

Any and all assistance would be appreciated greatly. And thanks for putting up with all of my questions :eek:
To set this up I will assume that we have a y coordinate system such that y = 0 ft at the point when the skydiver leaves the plane, so v = 0 ft/s both at t = 0 s, and I am setting the positive y direction to be downward.

As the diver is falling we have the following derivation of the equation of motion:
\(\displaystyle \sum F = m \frac{dv}{dt}\)

\(\displaystyle mg - 0.75v = m \frac{dv}{dt}\)
(mg is the weight and the friction force 0.75v points upward, the negative direction.)

So the equation is:
\(\displaystyle m \frac{dv}{dt} + 0.75v = mg\)

And of course distance is the time integral of the speed...

See what you can do with this and let us know.

-Dan
 
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  • #3
Thank you very very much for that!
 

Related to Skydiver/Physics Diff.Eq. Problem

What is the physics behind a skydiver's motion?

The motion of a skydiver is governed by the laws of physics, specifically the principles of gravity, air resistance, and Newton's laws of motion. As the skydiver falls, the force of gravity pulls them towards the ground, while air resistance pushes against them. This results in a constant acceleration towards the ground until the skydiver reaches terminal velocity.

How does air resistance affect a skydiver's speed?

Air resistance, also known as drag, is a force that opposes the motion of a skydiver. As the skydiver falls, air molecules collide with their body, creating a force that slows them down. This force increases as the skydiver's speed increases, until it eventually balances out with the force of gravity, resulting in a constant speed known as terminal velocity.

What is terminal velocity and how is it calculated?

Terminal velocity is the maximum speed that a falling object, such as a skydiver, can reach due to the balance of forces acting on it. It is calculated by setting the force of gravity equal to the force of air resistance and solving for the velocity. This can be expressed as: Vt = √(2mg/cdAρ), where m is the skydiver's mass, g is the acceleration due to gravity, cd is the drag coefficient, A is the cross-sectional area of the skydiver, and ρ is the density of air.

How does the mass of a skydiver affect their descent?

The mass of a skydiver affects their descent because it is a factor in calculating their terminal velocity. As the mass of a skydiver increases, so does their terminal velocity, meaning they will reach a higher constant speed during freefall. However, the force of air resistance also increases with mass, so a heavier skydiver will experience a greater amount of drag, resulting in a longer descent time.

What is the role of a parachute in a skydiver's descent?

A parachute is used to slow down and control a skydiver's descent. When a parachute is deployed, it creates a large surface area for air resistance to act upon, which significantly increases the drag force and slows down the skydiver's descent. This allows them to land safely at a slower speed without experiencing the impact of hitting the ground at terminal velocity.

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