Parachute and Air Resistance

[diracy]

Homework Statement

An ideal parachute accelerates all the air it sweeps as it travels, and the air is accelerated from
0 to the parachute's velocity. Using F = dp/dt calculate the force on an ideal parachute of area A as a function of its velocity and the density of air. Assuming you wouldn't break any limbs if you jumped from 5 feet high without a parachute, this tells you what steady-state velocity you would like your parachute to reach. What diameter a circular parachute you would need? Suppose you weigh 100 kg (including the parachute, a spare, and a video camera to document your jump) and the air density is 1.2 kg/m^3

Homework Equations

The Attempt at a Solution

I'm just a little confused about where to start with this one. Any help?

 

[rude man]

What is the change in momentum of the air molecules in 1 sec?

 

[diracy]

I'm not sure. I've been working on this one for a while, but I haven't gotten very far at all. Any help?

 

[diazona]

The question is "What diameter a circular parachute you would need?" So focus on the diameter of the parachute. What does that affect?

If you prefer, you could think about it a different way: The question says "Using F = dp/dt calculate the force on an ideal parachute..." So what do you need to know in order to calculate that force?

 

[paranormal]

I would approach this problem by first understanding the principles of air resistance and how it affects the motion of objects in a fluid medium like air. Air resistance is a force that acts in the opposite direction of an object's motion through air, and it increases as the speed of the object increases. This means that as the parachute falls through the air, it will experience an increasing force of air resistance that will eventually balance out the force of gravity pulling it downwards.

To calculate the force on the parachute, we can use the equation F = dp/dt, where F is the force, dp is the change in momentum, and dt is the change in time. In this case, the change in momentum is equal to the mass of the air swept by the parachute (ρAv) multiplied by the change in velocity (v). Therefore, the equation becomes F = ρAv^2, where ρ is the density of air, A is the area of the parachute, and v is the velocity.

To find the steady-state velocity that the parachute needs to reach in order to balance out the force of gravity, we can use the equation F = mg, where m is the mass of the parachute and g is the acceleration due to gravity. Solving for v, we get v = √(mg/ρA). Plugging in the given values of m (100 kg), g (9.8 m/s^2), and ρ (1.2 kg/m^3), we can calculate the desired velocity of the parachute.

To determine the diameter of the circular parachute needed, we can use the equation A = πr^2, where A is the area of the parachute and r is the radius. Solving for r, we get r = √(A/π). Plugging in the given values of A (which we can assume is equal to the area of the person plus parachute, which is roughly 2 m^2) and solving for r, we get a radius of approximately 0.8 meters. Therefore, the diameter of the parachute should be around 1.6 meters.

In conclusion, in order to safely jump from a 5-foot height without breaking any limbs, the parachute would need to reach a steady-state velocity of approximately 22 m/s and have a diameter of 1.6 meters. However, these calculations are based on ideal conditions and do not take into account other factors such as air turbulence or