Question

A balloon with mass 'm' is descending down with an acceleration 'a' (where a< g). How much mass should
be removed from it so that is starts moving up with an acceleration 'a'?

Answer 1

Hello!

Let F is the thrust of air on the balloon.

Then,
mg + F = ma

⇒ F = ma - mg

⇒ F = m(a - g)

Now,
Let m’ will be the mass when the balloon will fly upwards with acceleration ' a '.

In this case, the thrust will remain same!

Then,

- a = F / m’

⇒ - m’a = m(a - g)

⇒ m’ = - m(a - g) / a

Now, the mass we have to remove is :

m - m’

⇒ m + m(a - g) / a

⇒ m ( 1 + a - g / a )

⇒ m.(2a - g / a)

Cheers!

Answer 2

Given :-

Mass of balloon = m

There is action of Buoyant Force on the balloon which is acting in upward direction.

mg - B = ma ----------(1)

Here, we must assume that while removing some mass from the balloon its volume and buoyant force remains same.

Let the new mass of balloon = m'

Mass removed = m - m'

B - m'g = m'a -----(2)

From equation (1) & (2) we get,

mg - m'g = m'a + ma

mg - ma = m'g + m'a

m(g-a) = m'(g+a)

m' = m(g-a)/(g+a)

Again,

∆m = m - m'

∆m = m[1-(g-a)/(g+a)]

∆m = m[(g+a)-(g-a)/(g+a)]

∆m = m[g + a - g + a]/(g + a)

∆m = 2ma/(g+a)

Hence,

The mass can be removed = ∆m = 2ma/(g+a).