Question

A hot air balloon of total mass M (including passengers and luggage) is moving with a downward acceleration of magnitude a. As it approaches a mountain, the captain needs to accelerate upwards. He decides to throw enough ballast over board to achieve an upward acceleration of magnitude a/2. What fraction of the initial mass does he have to drop? Assume the upward lift force exerted by the air on the balloon does not change because of the decrease in mass.

Answer 1

The captain needs to drop the entire initial mass of the balloon to achieve the desired upward acceleration.

Step-by-step explanation:

To find the fraction of the initial mass the captain needs to drop, we can use Newton's second law of motion. We know that the downward acceleration of the hot air balloon is equal to the force of gravity divided by the total mass. So, the force of gravity is given by F = M * a, where M is the total mass and a is the downward acceleration.

When the captain throws enough ballast overboard to achieve an upward acceleration of a/2, the net force acting on the balloon becomes F = (M - x) * (a/2), where x is the mass dropped. Since the upward lift force exerted by the air on the balloon does not change, we can equate the two forces and solve for x.

Setting the two forces equal, (M - x) * (a/2) = M * a, we can solve for x:

x = M - 2M = -M

Since mass cannot be negative, we conclude that the captain needs to drop the entire initial mass of the balloon to achieve the desired upward acceleration.

Answer 2

The fraction of mass that was thrown out is calculated by the following Formula:

M - m = (3a/2)/(g²- (a²/2) - (ag/2))

Step-by-step explanation:

We know that Force on a moving object is equal to the product of its mass and acceleration given as:

F = ma

And there is gravitational force always acting on an object in the downward direction which is equal to g = 9.8 ms⁻²

Here as a convention we will use positive sign with acceleration to represent downward acceleration and negative sign with acceleration represent upward acceleration.

Case 1:

Hot balloon of mass = M

acceleration = a

Upward force due to hot air = F = constant

Gravitational force downwards = Mg

Net force on balloon is given as:

Ma = Gravitational force - Upward Force

Ma = Mg - F (balloon is moving downwards so Mg > F)

F = Mg - Ma

F = M (g-a)

M = F/(g-a)

Case 2:

After the ballast has thrown out,the new mass is m. The new acceleration is -a/2 in the upward direction:

Net Force is given as:

-m(a/2) = mg - F (Balloon is moving upwards so F > mg)

F = mg + m(a/2)

F = m(g + (a/2))

m = F/(g + (a/2))

Calculating the fraction of the initial mass dropped:

`M-m = (F)/(g-a) - (F)/(g+(a)/(2) )\\M-m = F*[(1)/(g-a) - (1)/(g+(a)/(2) )]\\M-m = F*[((g+(a/2)) - (g-a))/((g-a)(g+(a/2))) ]\\M-m = F*[(g+(a/2) - g + a))/((g-a)(g+(a/2))) ]\\M-m = F*[((3a/2))/(g^(2)-(a^(2))/(2)-(ag)/(2)) ]`