A hot-air balloon of mass 200 kg is descending vertically with downward acceleration of magnitude 1.2 m/s2. How much mass (ballast) must be thrown out to give the balloon an upward acceleration of magnitude 1.2 m/s2 (same magnitude but opposite direction)? Assume that the upward force from the air (the lift) does not change because of the decrease in mass.
Please don't cheat and use this website do your work and learn! - spread this message!
To determine the mass (ballast) that must be thrown out, we can 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 gravitational force pulling it downward and the upward force from the air (the lift).
Let's begin by finding the net force when the balloon is descending.
Step 1: Calculate the gravitational force pulling the balloon downward.
The gravitational force can be calculated using the formula:
Force = mass × acceleration due to gravity
Given:
Mass of balloon (m1) = 200 kg
Acceleration due to gravity (g) = 9.8 m/s^2
Gravitational Force = m1 × g
Gravitational Force = 200 kg × 9.8 m/s^2
Step 2: Calculate the net force when the balloon is descending.
The net force when descending is the sum of the gravitational force and the force due to acceleration in the downward direction.
Net Force descending = Gravitational Force - m1 × acceleration descending
Net Force descending = (200 kg × 9.8 m/s^2) - (200 kg × 1.2 m/s^2)
Now, let's find the mass (ballast) that must be thrown out to achieve upward acceleration of magnitude 1.2 m/s^2.
Step 3: Calculate the net force required when the balloon is ascending.
The net force required when ascending is the sum of the gravitational force and the force due to acceleration in the upward direction.
Net Force ascending = Gravitational Force + m2 × acceleration ascending
Net Force ascending = (200 kg × 9.8 m/s^2) + (m2 × (-1.2 m/s^2)) [Note: acceleration ascending is negative as it is in the opposite direction]
Since the magnitude of the upward acceleration is the same as the downward acceleration, we can set the net force descending equal to the net force ascending.
Net Force descending = Net Force ascending
(200 kg × 9.8 m/s^2) - (200 kg × 1.2 m/s^2) = (200 kg × 9.8 m/s^2) + (m2 × (-1.2 m/s^2))
Step 4: Solve for the mass (m2) of the ballast.
(200 kg × 9.8 m/s^2) - (200 kg × 1.2 m/s^2) = (200 kg × 9.8 m/s^2) + (m2 × (-1.2 m/s^2))
Now you can solve this equation for m2 to find the mass of the ballast that needs to be thrown out.
To find out how much mass (ballast) must be thrown out, we first need to determine the force required to achieve an upward acceleration of 1.2 m/s².
We can use Newton's second law, which states that the force (F) acting on an object is equal to the mass (m) of the object multiplied by its acceleration (a): F = m * a.
In this case, the acceleration (a) is -1.2 m/s² (negative because the acceleration is in the opposite direction), and we want to find the mass (m) that will give this acceleration. So, the equation becomes:
F = (-1.2) * m
The force required to achieve this acceleration is given by the difference in weight of the balloon before and after the ballast is thrown out.
Before throwing out the ballast, the weight of the balloon can be calculated as the product of its mass (m₁) and the acceleration due to gravity (g), where g is approximately 9.8 m/s².
Weight before = m₁ * g
After throwing out the ballast, the weight of the balloon is reduced because it has lost some of its mass (m₁ - m). So, the weight after throwing out the ballast can be calculated as:
Weight after = (m₁ - m) * g
Since the upward force from the air (the lift) remains constant regardless of the decrease in mass, the net force acting on the balloon is the difference between the weight before and the weight after:
F = Weight before - Weight after
F = m₁ * g - (m₁ - m) * g
Now we can equate the force required for the upward acceleration to the net force:
(-1.2) * m = m₁ * g - (m₁ - m) * g
Simplifying the equation:
-1.2m = m₁g - m₁g + mg
-1.2m = mg
1.2m = mg
Now canceling the m on both sides:
1.2 = g
Rearranging to solve for mass (m):
m = 1.2 / g
Substituting the value of g (approximately 9.8 m/s²):
m = 1.2 / 9.8
m ≈ 0.122 kg
Therefore, approximately 0.122 kg or 122 grams of mass (ballast) must be thrown out to give the balloon an upward acceleration of magnitude 1.2 m/s².
Let B = the buoyancy force and M = the initial total mass. Let m be the ballast mass released
M g - B = M*1.2 m/s^2
(M-m)g -B = -M*1.2 m/s^2
Solve for m.
m g = M*2.4m/s^2
m/M = 2.4/g
m = 0.245 M = 49 kg