A balloon carrying a basket is descendign with a constant acceleration of g/3. the total mass of the basket and contents is mB.you see water below and decide you need to throuw out some mass to allow the basket to rise and not hit the water. how much mass should you throw out to allow the ballon-basket to rise witha an acceleration of g/2? the only force to consider other than the weight is the upward directed buoyancy force, fB due to the hot air.
To find out how much mass you should throw out to allow the balloon-basket to rise with an acceleration of g/2, we need to consider the forces acting on the system.
1. Weight (mg): The weight of the balloon-basket and its contents acts downward with a force equal to the mass (m) multiplied by the acceleration due to gravity (g).
2. Buoyancy force (fB): The upward buoyancy force is due to the displaced air. It acts opposite to the weight and can be calculated as the weight of the air displaced by the balloon.
Given that the balloon-basket is descending with a constant acceleration of g/3, the net force acting on it is:
Net force = Weight - Buoyancy force
Now, we want the balloon-basket to rise with an acceleration of g/2. This means the net force acting on it should be in the upward direction. Therefore, we can set up the following equation:
Net force = Weight - Buoyancy force = mass * (g/2)
Substituting the weight and buoyancy force:
mg - fB = m * (g/2)
Now, let's express the buoyancy force (fB) in terms of the weight of the displaced air:
fB = mB * g
Substituting this value back into the equation:
mg - mB * g = m * (g/2)
Simplifying the equation:
g(m - mB) = g(m/2)
Dividing both sides by g:
m - mB = m/2
To isolate the mass (m) that should be thrown out, we can rearrange the equation:
m = 2mB
So, in order to allow the balloon-basket to rise with an acceleration of g/2, you should throw out a mass equal to twice the total mass of the basket and its contents (2mB).
To determine how much mass you should throw out to allow the balloon-basket to rise with an acceleration of g/2, we need to consider the forces involved.
Let's break down the forces acting on the balloon-basket system:
1. Weight (downward force): The weight of the balloon and the contents is given by W = mB * g/3, since the acceleration is g/3.
2. Buoyancy (upward force): The buoyancy force, fB, acts upward and opposes the weight. Since the balloon is filled with hot air, we can assume that it displaces an equal volume of air, resulting in a buoyancy force of fB = mB * g.
3. Net force: The net force is the difference between the weight and the buoyancy force. The net force required to accelerate the system upwards with an acceleration of g/2 is given by Fnet = mB * (g/2 - g/3).
To find the mass that should be thrown out, we need to determine the change in mass, Δm.
Since the net force is equal to the mass times the acceleration, we can set it equal to the net force expression from above:
Fnet = mB * (g/2 - g/3)
mB * (g/2 - g/3) = Δm * (g/3)
Simplifying and solving for Δm, we get:
Δm = mB * (g/2 - g/3) / (g/3)
Δm = mB * ((3g - 2g) / (2g)) / (g/3)
Δm = mB * (g / (2g)) / (g/3)
Δm = mB * (3/2)
Therefore, you should throw out (3/2) times the mass of the basket and contents (mB) to allow the balloon-basket to rise with an acceleration of g/2.