Several people are riding in a hot-air balloon. The combined mass of the people and balloon is 328 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 0.18 m/s2?
To answer this question, we need to understand the forces acting on the hot-air balloon.
The downward-acting force on the balloon is the weight of the people and the balloon itself, which is equal to the combined mass (328 kg) multiplied by the acceleration due to gravity (9.8 m/s^2).
The upward-acting force is the buoyant force, which is equal to the weight of the air displaced by the balloon. Since the balloon is motionless, the buoyant force is currently equal to the weight of the people and the balloon.
To make the balloon acquire an upward acceleration, the upward-acting force (buoyant force) needs to be greater than the downward-acting force (weight).
To calculate the downward-acting force, we use the formula:
Downward-acting force = mass * gravitational acceleration
Downward-acting force = 328 kg * 9.8 m/s^2 = 3214.4 N
To find the upward-acting force, we can use the formula:
Upward-acting force = mass * upward acceleration
Since the mass of the balloon and the people remains constant, the m in this formula represents the dropped mass overboard.
Now we need to solve for the dropped mass:
Upward-acting force = (328 kg - dropped mass) * 0.18 m/s^2
We can rearrange the formula to solve for the dropped mass:
dropped mass = (328 kg * 0.18 m/s^2) / 0.18 m/s^2
dropped mass = 328 kg - 328 kg * 0.18 m/s^2 / 0.18 m/s^2
dropped mass = 328 kg - 328 kg
dropped mass = 0 kg
Therefore, no mass needs to be dropped overboard for the balloon to acquire an upward acceleration of 0.18 m/s^2, as the combined mass of the balloon and people remains constant.
To solve this problem, we need to use Newton's second law of motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration.
In this case, the net force acting on the balloon is the difference between the upward buoyant force and the downward weight force. When the balloon is motionless, these forces are balanced, so their magnitudes are equal. However, when the balloon acquires an upward acceleration, the buoyant force becomes greater than the weight force.
Let's denote the mass of the balloon as M, the mass of the people as m, and the mass that needs to be dropped overboard as x.
The weight force acting on the balloon and people is given by F_weight = (M + m) * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
The buoyant force is given by F_buoyant = M * g.
To calculate the net force acting on the balloon, we can subtract the weight force from the buoyant force:
Net Force = F_buoyant - F_weight
= M * g - (M + m) * g
= M * g - M * g - m * g
= - m * g
Since acceleration is defined as the net force divided by the mass, we have:
a = Net Force / (M + m)
- 0.18 m/s^2 = (- m * g) / (M + m)
Now we can plug in the known values:
0.18 m/s^2 = (- x * g) / (M + m)
To solve for x, we can rearrange the equation:
x = - (0.18 m/s^2) * (M + m) / g
Finally, we substitute the given values:
x = - (0.18 m/s^2) * (328 kg + m) / (9.8 m/s^2)
Note: The negative sign indicates that mass needs to be dropped overboard.
So, to find the value of x, we need to know the mass of the people, m.