Question) A balloon of mass “m” is moving vertically downward with an accelaration “a” how much mass removed from it so that it will move vertically upward with an accelaration ?

To determine how much mass needs to be removed from a balloon so that it can change its motion from moving downward to moving upward with a specific acceleration, we can apply Newton's second law of motion and the principles of buoyancy. Let's break this down step by step.

Understanding Forces Acting on the Balloon

When the balloon is moving downward with an acceleration "a," several forces are at play:

• Weight of the Balloon (W): This is the force due to gravity acting on the balloon, calculated as W = mg, where "g" is the acceleration due to gravity (approximately 9.81 m/s²).
• Buoyant Force (B): This is the upward force exerted by the fluid (air, in this case) on the balloon. According to Archimedes' principle, the buoyant force is equal to the weight of the fluid displaced by the balloon.

Setting Up the Equation

When the balloon is descending with an acceleration "a," the net force acting on it can be expressed as:

Net Force (F_net) = Weight - Buoyant Force = ma

Rearranging this gives us:

mg - B = ma

Condition for Ascending Motion

For the balloon to ascend with an acceleration "a' (which we want to be positive), the net force must be in the opposite direction:

Net Force (F_net) = Buoyant Force - Weight = ma'

Rearranging this gives us:

B - mg' = ma'

Combining the Equations

To find the mass that needs to be removed, we need to set the buoyant force equal to the new weight of the balloon when some mass is removed:

B = m' g

Where "m'" is the new mass of the balloon after removing some mass "Δm." Thus, we can express:

m' = m - Δm

Final Equation

Now we can combine our equations. From the descending condition:

mg - B = ma

And for ascending:

B - (m - Δm)g = ma'

Setting these equal gives us:

mg - (m - Δm)g = ma + ma'

Rearranging leads to:

Δmg = ma + ma'

From this, we can solve for the mass that needs to be removed:

Δm = (ma + ma') / g

Conclusion

In summary, to make the balloon ascend with an acceleration "a'," you need to remove a mass Δm calculated by the formula:

Δm = (ma + ma') / g

This formula allows you to determine how much mass to remove based on the initial downward acceleration "a" and the desired upward acceleration "a'." By applying these principles, you can effectively manipulate the balloon's motion in a controlled manner.