How fast does the balloon accelerate upwards?

[osker246]

Homework Statement

your team is in charge of launching a large helium weather balloon that is spherical in shape, and whose radius is 3.0 m and total mass is 17.5 kg (balloon plus helium plus equipment).

(a) What is the initial upward acceleration of the balloon when it is released from sea level?
(b) If the drag force on the balloon is given by the equation below, where r is the balloon radius, ρ is the density of air, and v is the balloon's ascension speed, calculate the terminal velocity of the ascending balloon.

Homework Equations

V=(4/3)Pi*r3
F=ma
density of air = 1.29 Kg/m^3
density of He = 0.1785 Kg/m^3
g= 9.81 N/Kg

The Attempt at a Solution

I am attempting to do this online homwork and the first hint they give is to draw a FB diagram of the three forces acting on object. From what I think i know the 3 forces are...

F_{air pushing up on balloon +} F_{Buoyancy balloon} -W_balloon=ma

is this correct?

If so I know the Weight=mg, so that's done. But I am not too positive about F_{air pushing up on balloon +} F_{Buoyancy balloon}. Now does F_{air pushing up on balloon= Vballoon*densityair*g? Also assuming the volume of the sphere is completely filled with helium, does} F_{Buoyancy balloon}=V_He*Density_He*g?

If somebody could help me out I would greatly appreciate it. I am not quite sure if I know what I am doing here...

 

[yinx]

Archimedes principle: when a body(balloon in this case) is submerged in a fluid (air in this case) the fluid exerts an upward force on the body equals to the weight of the fluid displaced by the body.

try understanding the Archimedes principle and work towards it.

 

[osker246]

Hi yinx,

Thanks for the reply, I did a little more studying on archimedes principle and the concept does make more sense. So I was able to figure out the answer to part a. Turns out the formula was F_b-W_b=ma, where F_b=rho_air*V*g. The answer ended up being 72 m/s^2.

I also forgot to include an equation that was given for part B. F_d=(1/2)(pi^2)(rho)V^2.

So from what I can tell the third force is drag force. So now the drag force would be pushing down on the balloon right? So F_d=F_b - W_b - ma? So from the solved value of drag force I can use the equation provided above to solve for the accent velocity? This is the only way I can think of yet I get a answer no where close to the correct answer. Any ideas?

 

[osker246]

Ok, so I figured out my mistake I forgot to incorporate r^2 Fd=(1/2)(pi^2)(rho)V^2. But one thing I don't understand from what I read online from a forum is drag force is buoyant force - Weight. I don't understand this and my textbook mentions nothing of this. Does anybody know of some resources that could explain this?

 

[rwisz]

I can say that the initial upward acceleration of the balloon can be calculated using the equation F=ma, where F is the net force acting on the balloon, m is its mass, and a is the acceleration. The net force can be found by subtracting the weight of the balloon from the sum of the buoyant force and the force of air resistance. This can be represented as follows:
a=(FBuoyancy-Fair-Wballoon)/m

To calculate the terminal velocity of the ascending balloon, we can use the equation for drag force, F=1/2*ρ*v^2*Cd*A, where ρ is the density of air, v is the velocity of the balloon, Cd is the drag coefficient, and A is the cross-sectional area of the balloon. We can set this equal to the weight of the balloon (mg) and solve for v. This will give us the velocity at which the drag force will balance the weight of the balloon, resulting in a constant velocity or terminal velocity.
v=sqrt(2mg/(ρ*Cd*A))

It is important to note that Cd and A will vary depending on the shape and size of the balloon, as well as the properties of the surrounding air. Therefore, to get a more accurate value for the terminal velocity, we would need to know the specific values for these variables.

In conclusion, the initial upward acceleration of the balloon can be calculated using the net force equation, and the terminal velocity can be calculated using the drag force equation. However, to get more accurate results, we would need to know more specific information about the balloon and its surroundings.