2. A hot-air balloon experiences an acceleration of 1.10 m/s2 [down]. The total
mass of the balloon, the basket, and the contents of the basket is 315 kg.
(a) What is the upward (buoyant) force on the system?
(b) The balloonist wishes to change the acceleration to zero. There is no fuel
left to heat the air in the balloon. Determine the mass of the ballast that
must be discarded overboard. (Neglect air resistance.)
(a) To find the upward buoyant force, we first need to determine the total weight of the system. Weight is calculated using the formula:
Weight = mass x acceleration due to gravity
Weight = 315 kg x 9.81 m/s^2
Weight = 3091.5 N
Since the hot-air balloon is experiencing an acceleration of 1.10 m/s^2 downward, the net force acting on the system is the sum of the weight and the buoyant force. The buoyant force is equal in magnitude but opposite in direction to the weight:
Buoyant force = Weight - mass x acceleration
Buoyant force = 3091.5 N - 315 kg x 1.10 m/s^2
Buoyant force = 3091.5 N - 346.5 N
Buoyant force = 2745 N
Therefore, the upward (buoyant) force on the system is 2745 N.
(b) To change the acceleration to zero, the net force on the system must be zero. The only force that can be adjusted is the weight, by discarding ballast overboard. Let the mass of the discarded ballast be denoted by "m."
Weight = (315 kg - m) x 9.81 m/s^2
Weight = 3091.5 N - 9.81m
Since the net force is 0 when acceleration is zero, the buoyant force must equal the weight:
Buoyant force = Weight
2745 N = 3091.5 N - 9.81m
9.81m = 346.5 N
m = 35.3 kg
Therefore, the mass of the ballast that must be discarded overboard is 35.3 kg.