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.
Well, it sounds like this hot-air balloon is having a bit of a weight problem, huh? Don't worry, I'm here to lighten the mood and help you out!
To figure out how much mass needs to be thrown out, we need to find the difference in the downward force and the desired upward force.
According to Newton's second law, the force acting on an object is equal to its mass multiplied by its acceleration. In this case, the downward force is given by F = m * a (where m is the mass of the balloon and a is the downward acceleration of 1.2 m/s^2).
To calculate the desired upward force, we'll use the same formula: F = m * a. However, this time the acceleration is -1.2 m/s^2 (since the upward acceleration is in the opposite direction).
Now, to find the difference between the two forces (F_upward - F_downward), we subtract the two equations:
m * (-1.2) - m * 1.2 = 0
Simplifying the equation, we get:
-2.4m = 0
Dividing both sides by -2.4, we find:
m = 0 kg
Well, ain't that a bit of a conundrum?! It seems like no mass needs to be thrown out in order to achieve the desired upward acceleration. Looks like this balloon is just a natural born climber, no diet necessary!
To find out how much mass (ballast) must be thrown out, we can use Newton's second law of motion which states that the force on an object is equal to its mass multiplied by its acceleration.
Given:
- Mass of the hot-air balloon = 200 kg
- Downward acceleration = 1.2 m/s^2 (initial)
- Upward acceleration = 1.2 m/s^2 (final)
Let's assume the mass of the ballast to be thrown out is "m" kg.
Step 1: Calculate the downward force on the balloon (initial situation):
Force = Mass × Acceleration
Force_initial = 200 kg × 1.2 m/s^2
Force_initial = 240 N
Since the downward force is equal to the weight of the balloon including the ballast, we can calculate the weight as:
Weight_initial = Mass × Acceleration due to gravity
Weight_initial = 200 kg × 9.8 m/s^2
Weight_initial = 1960 N
Therefore, the weight of the balloon is 1960 N.
Step 2: Calculate the upward force required on the balloon (final situation):
Force_final = Mass × Acceleration
Force_final = (200 - m) kg × 1.2 m/s^2
Force_final = (240 - 1.2m) N
Step 3: Equate the forces in the final and initial situations to find the value of "m":
Force_initial = Force_final
240 N = 240 N - 1.2m N
Simplifying the equation:
1.2m N = 0 N
m = 0 kg
Therefore, no mass (ballast) needs to be thrown out to give the balloon an upward acceleration of 1.2 m/s^2.
To solve this problem, we need to consider the forces acting on the hot-air balloon during both descension and ascension.
When the balloon is descending, the forces acting on it are the weight due to gravity (mg) and the buoyant force (FB) exerted by the air. The net force is the difference between these two forces and is equal to the mass of the balloon multiplied by its downward acceleration (ma).
So, the net force when descending is given by:
Net force = mg - FB = ma
When the balloon is ascending, its acceleration is in the opposite direction, so the net force is:
Net force = FB - mg = ma'
where a' is the upward acceleration we want to achieve.
Since the buoyant force (FB) remains the same during both descension and ascension, we can equate the two expressions for the net force:
ma = FB - mg
ma' = FB - mg
Setting the expressions equal to each other:
ma = ma'
ma - ma' = FB - mg
(m - m)a' = FB - mg
a' = (FB - mg)/(m - m)
Rearranging the equation, we get:
a' = (FB - mg)/(m - m)
m - m = (FB - mg)/a'
Now, rearrange the equation to solve for m, the mass to be thrown out:
m = (FB - mg)/a' + m
Substituting the given values:
m = (FB - (200 kg)(9.8 m/s^2))/(1.2 m/s^2) + 200 kg
Now, we need to know the value of the buoyant force (FB) to calculate the mass to be thrown out. The buoyant force can be calculated using Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object.
However, the problem statement assumes that the buoyant force remains constant, so we can omit FB from our calculations.
m = (200 kg)(9.8 m/s^2)/(1.2 m/s^2) + 200 kg
m = 1633.33 kg + 200 kg
m ≈ 1833.33 kg
Therefore, approximately 1833.33 kg of mass (ballast) must be thrown out to give the balloon an upward acceleration of magnitude 1.2 m/s^2.