What Parachute Size Ensures a Safe Ambulance Airdrop?

[sirvantehs]

Homework Statement

An ambulance needs to be delivered to a remote town devastated by a major earthquake. All roads leading into the town are blocked due to the earthquake and the ambulance can only be rushed to the area by airlift. The ambulance will be pushed out of a military cargo jet at 3000 m altitude
and rescue staff need to find out what kind of parachute is needed for this mission. The drag force is given by the approximate formula: F = ¼ ρAv2, where ρ is the density of air and ρ = 1.2 kg/m3, A is the area of the cross-section of the parachute perpendicular to the motion and v is the
velocity.

What should the diameter of the parachute be so that the ambulance can land safely?

Notice that you have to make assumptions on what is approximately a safe landing velocity and what is a weight of a typical ambulance.

Make sure you justify all your assumptions.

Homework Equations

mg = (1/4)pAv^2

The Attempt at a Solution

I isolated A (area):
A = (4mg)/(pv^2)

I replaced A with the formula for the area of a circle:
(1/4)∏d^2 = (4mg)/(pv^2)

I isolated d (diameter):
d = √(16mg)/(1.2∏v^2)

Assuming the ambulance weighs a relatively low 3500kg, and that the ambulance lands at 6 m/s (airbags are typically deployed at around a collision speed of 6.2 m/s), I get a parachute diameter of approximately 63.6 m

The problem is my teacher says that value is far, far too big. What am I missing?

Thank you for your time.

 

[Simon Bridge]

Consider the speed that air bags deploy - would that speed, in collision with a stationary object, damage the vehicle?

 

[sirvantehs]

I definitely feel like that speed would damage the vehicle. I suppose I'm trying to contextualize the speed in terms of something specific, like the airbags going off. I was hoping that, by being shy of the airbag deployment speed, I wouldn't absolutely wreck the ambulance. I suppose I'm trying to find the highest possible "safe" speed in order to minimize the diameter of the parachute.

I feel like, in reality, I would need to slow down the ambulance's descent, but I don't know how to (essentially) over-compensate for that, in order to reduce the parachute's diameter.

 

[paisiello2]

Why on Earth they need to air drop in an ambulance, when they can just air drop in any medical equipment they need directly to the spot in question, is beyond me. But let's ignore that for the moment.

Given the restrictions of the problem the only things I can think of to reduce the diameter:

1) largest diameter parachute ever made is 150 feet (45m) proposed for use in returning rocket boosters, so maybe you can propose to use 2 or 3 smaller diameter parachutes at the same time.

2) include the drag on the ambulance itself which is not inconsiderable, but a large parachute itself might also weigh a lot so you should include this weight also

3) does the ambulance have to be dropped from a large enough height to reach it's terminal velocity? as long as it is dropped from some height high enough to fully deploy the parachute then they could fly in lower to reduce the speed

4) now the problem assumed the parachute was round but why couldn't you use a rectangular airfoil shape and glide the ambulance in? this would create a lift to reduce the vertical speed but would cause some amount of horizontal speed requiring a runway which might be prohibitive in this case

 

[haruspex]

Consider the speed that air bags deploy - would that speed, in collision with a stationary object, damage the vehicle?

You seem to be suggesting a lower speed, but that would make the parachute even bigger, and the teacher is reported to have said the answer is already too great.
Presumably the vehicle lands on its wheels, so the maximum speed is determined by the suspension system. Do vertical accelerations trigger airbags? - don't know. 6 m/s would be like dropping from 1.8 m. Sounds like quite a challenge to the suspension, so I agree that the speed may be already too high.
The given formula for drag corresponds to a coefficient of 0.5. That is the value for a smooth sphere in turbulent flow, so it seems a bit low to me. On the net I see values from 0.75 to 1.5, dependent on parachute design.
I see no errors in the calculations in the OP. Maybe there are some lighter ambulances available.

 

[AlephZero]

The military do airdrop vehicles like humvees and even battle tanks, but if you look at some youtube videos of airdrops, a 60m diameter chute is obviously too big.

The OP's drag coefficient for the parachutes is too low and there is also the drag from the object being dropped. But that doesn't seem enough to get to the right ballpark for the diameter.

I couldn't find a video which gave a good view of the actual landing. That's probably not too surprising, considering you don't want to be standing with a camera where a hummer might land on your head, or throw up a shower of stones when it hits the ground, or you get whiplashed by a parachute cable, or whatever.

 

[AlephZero]

Another line of attack is to Google for "airdrop parachutes". You can find some technical data from the suppliers, but the OP can find that for him/herself. That gives you a "right" answer to aim for, even if it doesn't tell you how to do the calculation.

 

[Simon Bridge]

You seem to be suggesting a lower speed, but that would make the parachute even bigger, and the teacher is reported to have said the answer is already too great.

... yeah: I misread the result meaning the speed was too great.

64m does seem on the big side.
Maybe the issue is with the model used?
http://www.pcprg.com/rounddes.htm

Another thing to consider is if the ambulance is being dropped onto it's suspension, maybe it can withstand a higher speed landing? This would involve figuring how much energy the suspension can safely absorb.