Several people are riding in a hot-air balloon. The combined mass of the people and balloon is 308 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 0.11 m/s2?

To determine how much mass should be dropped overboard so the balloon acquires an upward acceleration of 0.11 m/s^2, we need to use Newton's second law of motion.

1. First, let's calculate the net force acting on the balloon when it acquires an upward acceleration of 0.11 m/s^2.

Net force = mass × acceleration

Since the balloon and people have a combined mass of 308 kg, the net force can be calculated as follows:

Net force = (308 kg) × (0.11 m/s^2)

2. The net force acting on the balloon is equal to the difference between the buoyant force and the weight of the balloon and people, so we can write the equation as:

Net force = Buoyant force - Weight

Rearranging the equation, we have:

Buoyant force = Net force + Weight

Substituting the values, the equation becomes:

Buoyant force = (308 kg) × (0.11 m/s^2) + Weight

3. The weight of an object is given by the equation:

Weight = mass × gravitational acceleration

Assuming the gravitational acceleration is approximately 9.8 m/s^2, the weight of the balloon and people is calculated as:

Weight = (308 kg) × (9.8 m/s^2)

4. Substituting the weight value back into the buoyant force equation, we have:

Buoyant force = (308 kg) × (0.11 m/s^2) + (308 kg) × (9.8 m/s^2)

5. Simplifying the equation:

Buoyant force = (33.88 kg·m/s^2) + (3018.4 kg·m/s^2)

Buoyant force = 3052.28 kg·m/s^2

6. Finally, we need to calculate the mass that should be dropped overboard so the balloon acquires an upward acceleration of 0.11 m/s^2.

Buoyant force = mass dropped × gravitational acceleration

Rearranging the equation, we have:

mass dropped = Buoyant force / gravitational acceleration

Substituting the values:

mass dropped = (3052.28 kg·m/s^2) / (9.8 m/s^2)

mass dropped ≈ 311.00 kg

Therefore, approximately 311 kg of mass should be dropped overboard so the balloon acquires an upward acceleration of 0.11 m/s^2.

To find the mass that needs to be dropped overboard for the balloon to acquire an upward acceleration, we can use Newton's second law of motion:

F = m * a

Where F is the net force acting on the balloon, m is the mass of the balloon and people, and a is the acceleration.

In this case, the net force is equal to the difference between the buoyant force acting upward and the weight acting downward:

F = F_buoyant - F_weight

Since the balloon is motionless initially, the buoyant force is equal to the weight:

F_buoyant = F_weight

So, we can rewrite the equation as:

F = 0

Since the buoyant force remains constant, the only force acting downwards is the weight. Thus, the weight can be represented as:

F_weight = m * g

Where g is the acceleration due to gravity (approximately 9.8 m/s^2).

Now, we can rewrite the equation using all the given information:

0 = m * a - m * g

To find the mass that needs to be dropped, we need to rearrange the equation to solve for m:

m * a = m * g
m * (a - g) = 0
m = 0 / (a - g)
m = 0

From the equation, we can see that the mass that needs to be dropped in order to acquire an upward acceleration is zero. This means that no mass needs to be dropped overboard to achieve the desired acceleration of 0.11 m/s^2, as the buoyant force is equal to the weight.

WEll, the net force upward will be the weight of the mass dropped.

F=ma
mass*g=(308-mass)*acceleration

solve for mass dropped