A balloon of total mass 840 kg is rising vertically with constant speed. As a result of releasing some ballast, the balloon immediately accelerates at 0.5 m/s². Calculate the mass of the ballast released.
Firstly, when the balloon is moving uniformly,
the buoyant force F is equal to the gravity
F = m•g.
After the ballast releasing,
the buoyant force F isn’t changed
because it is equal to the weight of the air
displaced by the balloon.
Now the equation of the balloon motion is
(m-Δm) •a = F – (m - Δm) •g = m•g– (m - Δm) •g,
Δm = m•a/g-a = 840•0.5/(9.8-0.5) = 45.16 kg.
To solve this problem, we can use Newton's second law of motion, which states that the force acting on an object is equal to the product of its mass and acceleration.
In this case, the force acting on the balloon is due to the difference between the force of gravity and the buoyant force. When the balloon is initially rising at constant speed, these forces are balanced, so we have:
Force of gravity = Buoyant force
To find the mass of the ballast released, we need to determine the difference in the forces acting on the balloon before and after releasing the ballast.
Let's denote the mass of the ballast released as "m".
Before releasing the ballast:
Force of gravity = mass of the balloon * acceleration due to gravity = 840 kg * 9.8 m/s²
After releasing the ballast, the acceleration changes to 0.5 m/s². The force of gravity remains the same, but the buoyant force increases as the weight of the balloon decreases.
Buoyant force after releasing the ballast:
Buoyant force = (mass of the balloon - mass of the ballast) * acceleration due to gravity = (840 kg - m) * 9.8 m/s²
Setting the two forces equal:
840 kg * 9.8 m/s² = (840 kg - m) * 9.8 m/s²
Simplifying the equation:
840 kg = 840 kg - m
Rearranging the equation:
m = 840 kg - 840 kg
m = 0 kg
Therefore, the mass of the ballast released is 0 kg.
To calculate the mass of the ballast released, we need to make use of Newton's second law of motion, which states that the force acting on an object is equal to its mass multiplied by its acceleration (F = ma).
In this case, the force acting on the balloon is the gravitational force pulling it downwards, which can be calculated using the equation F = mg, where m is the mass of the balloon and g is the acceleration due to gravity (approximately 9.8 m/s²).
Since the balloon is rising vertically with constant speed, the net force acting on it must be zero. This means that the force due to the ballast released must be equal in magnitude and opposite in direction to the gravitational force:
F_ballast = -F_gravity
m_ballast * a = -m_balloon * g
Now, we can substitute the given values into the equation and solve for m_ballast:
m_ballast * 0.5 m/s² = -840 kg * 9.8 m/s²
m_ballast = (-840 kg * 9.8 m/s²) / 0.5 m/s²
m_ballast ≈ 16608 kg
Therefore, the mass of the ballast released is approximately 16608 kg.