Several people are riding in a hot-air balloon. The combined mass of the people and balloon is 310 kg. The balloon is motionless in the air, because the downward-acting weight of the people and balloon is balanced by an upward-acting “buoyant” force. If the buoyant force remains constant, how much mass should be dropped overboard so the balloon acquires an upward acceleration of 0.15 m/s2?
To solve this problem, we need to apply Newton's second law of motion, which states that the total force acting on an object is equal to the product of its mass and acceleration.
Step 1: Determine the net force acting on the balloon.
The net force acting on the balloon is the difference between the buoyant force and the weight of the people and balloon. Since the given information states that the balloon is motionless, we can infer that these two forces are balanced and the net force is zero.
Step 2: Calculate the weight of the people and balloon.
Weight is calculated using the formula: weight = mass × gravitational acceleration (W = m × g, where g is approximately 9.8 m/s^2)
Given that the combined mass of the people and balloon is 310 kg, the weight can be calculated by multiplying this mass by the acceleration due to gravity:
Weight = 310 kg × 9.8 m/s^2
Step 3: Calculate the required net force for the upward acceleration.
The balloon needs to have an upward acceleration of 0.15 m/s^2. Therefore, the net force required can be calculated using the formula:
Net force = mass × acceleration
For the upward direction, the net force is equal to the weight of the people and balloon plus the buoyant force. Since the net force is zero when the balloon is motionless, we need to add the buoyant force to the weight to get the required net force for upward acceleration:
Net force = Weight + Buoyant force
Step 4: Calculate the buoyant force.
The buoyant force is given by the formula: buoyant force = density × volume × gravitational acceleration.
Since the buoyant force remains constant, the density and gravitational acceleration do not change. Thus, we can set up an equation showing the relationship between the buoyant force, weight, and required net force:
Buoyant force = Net force - Weight
Step 5: Calculate the mass required to be dropped overboard.
To calculate the mass required to be dropped overboard, we need to find the decrease in weight caused by this mass. Using the weight formula W = m × g, we can rearrange it to solve for the mass:
Weight = mass × gravitational acceleration
mass = Weight / gravitational acceleration
Now we can proceed with the calculations using the given values.
To solve this problem, we need to apply Newton's second law, which states that the net force acting on an object is equal to the product of its mass and acceleration.
In this case, the net force on the balloon is the difference between the upward buoyant force and the downward weight of the people and balloon. Since the buoyant force remains constant, the change in net force will be caused by changing the weight.
Step 1: Calculate the weight of the balloon and people
The weight (W) can be calculated using the formula W = mass x gravity, where gravity is approximately 9.8 m/s^2.
W = 310 kg x 9.8 m/s^2 = 3,038 N
Step 2: Calculate the net force
The net force (F_net) is the difference between the buoyant force (F_buoyant) and the weight (W).
F_net = F_buoyant - W
Step 3: Calculate the mass to be dropped
To find the mass that needs to be dropped to achieve the desired acceleration, we rearrange the formula F_net = mass x acceleration and solve for mass.
mass = F_net / acceleration
mass = F_buoyant - W / acceleration
mass = 0.15 m/s^2 / acceleration
Step 4: Plug in the numbers and solve
mass = (0.15 m/s^2) / (9.8 m/s^2) = 0.015 kg
Therefore, to acquire an upward acceleration of 0.15 m/s^2, approximately 0.015 kg of mass should be dropped overboard.
The decrease in weight equals the new net accelerating force.
(Dropped mass)* g = M a
Solve for the dropped mass.
The acceleration will not continue forever.. Eventually aerodynamic drag limits the speed.