A hot-air balloon is accelerating upward under the influence of two forces, its weight and the buoyant force. For simplicity, consider the weight to be only that of the hot air within the balloon, thus ignoring the balloon fabric and the basket. The hot air inside the balloon has a density of ñhot air = 0.93 kg/m3, and the density of the cool air outside is ñcool air = 1.29 kg/m3. What is the acceleration of the rising balloon?
Ignore mass of balloon and basket.
Consider 1 m^3 of hot air:
mass, m = 0.93 kg
net upward force, F
=(1.29-0.93)=0.36 kg
= 9.81*0.36 N
=3.53 N
acceleration
= F/m
= 3.53/0.93 m/s²
= 3.8 m/s²
Note: At this acceleration, air resistance is significant.
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To find the acceleration of the rising balloon, we can use the difference between the weight of the hot air and the buoyant force acting on the balloon.
The weight of the hot air can be calculated using the formula:
Weight = Mass x Gravity
The mass of the hot air can be calculated using the formula:
Mass = Density x Volume
The volume of the hot air can be calculated using the formula:
Volume = (Mass of hot air) / (Density of hot air)
Since the weight of the hot air is equal to the buoyant force, we can write the equation:
Weight = Buoyant force
Now let's calculate the weight of the hot air. Since we are ignoring the weight of the balloon fabric and the basket, we can assume that the mass of the hot air is equal to the mass of the air displaced by the balloon:
Mass = Volume x Density
where the density is the density of the cool air outside.
From the given information, we have:
Density of cool air (ñcool air) = 1.29 kg/m3
Density of hot air (ñhot air) = 0.93 kg/m3
Now we can proceed with the calculations:
Mass = Volume x Density
= (Mass of hot air) / (Density of hot air)
= (Mass of hot air) / (ñhot air)
= (Weight of hot air) / (Gravity) [Using Weight = Mass x Gravity]
Since the weight of the hot air is equal to the buoyant force, we can write:
Buoyant force = Weight of hot air = Mass x Gravity
Now substitute the values:
Buoyant force = (Weight of hot air) = (Mass of hot air) x Gravity
= (Volume x Density of cool air) x Gravity
= (Density / Density of hot air) x Gravity
Simplifying, we have:
Buoyant force = (Density of cool air / Density of hot air) x Gravity
Now, solve for acceleration:
Weight of hot air = Buoyant force
Mass x Gravity = (Density of cool air / Density of hot air) x Gravity
Since Gravity is common on both sides, it cancels out:
Mass = (Density of cool air / Density of hot air) x Gravity
Now substitute the values:
Mass = (1.29 kg/m3 / 0.93 kg/m3) x 9.8 m/s2
Simplifying, we have:
Mass = 1.39 x 9.8 m/s2
Mass = 13.622 kg
Now we can calculate acceleration:
Acceleration = (Weight of hot air - Buoyant force) / Mass
Acceleration = (Mass x Gravity - Mass x Gravity) / Mass
Acceleration = 0 / Mass
Therefore, the acceleration of the rising balloon is 0 m/s2.
To find the acceleration of the rising balloon, we need to determine the net force acting on it.
First, let's calculate the weight force acting on the hot air inside the balloon. Weight is given by the formula:
Weight = mass * acceleration due to gravity
The mass of the hot air inside the balloon can be calculated using the formula:
Mass = density * volume
Given that the density of the hot air inside the balloon is 0.93 kg/m^3, and assuming the volume of the hot air inside the balloon is V, we can write:
Mass = 0.93 kg/m^3 * V
Now, let's find the buoyant force acting on the balloon. The buoyant force is given by the formula:
Buoyant force = weight of the fluid displaced = density of fluid * volume * acceleration due to gravity
The fluid being displaced is the cool air outside the balloon, so we use the density of cool air (1.29 kg/m^3) and the same volume V:
Buoyant force = 1.29 kg/m^3 * V * g
where g is the acceleration due to gravity, approximately 9.8 m/s^2.
The net force acting on the balloon is the difference between the buoyant force and the weight force:
Net force = Buoyant force - Weight
Net force = (1.29 kg/m^3 * V * g) - (0.93 kg/m^3 * V * g)
Simplifying further:
Net force = (1.29 - 0.93) kg/m^3 * V * g
Net force = 0.36 kg/m^3 * V * g
Now, we know that the net force is equal to mass times acceleration (Newton's second law):
Net force = mass * acceleration
Equating the two expressions for the net force:
0.36 kg/m^3 * V * g = (0.93 kg/m^3 * V) * acceleration
The volume V cancels out, so we are left with:
0.36 kg/m^3 * g = 0.93 kg/m^3 * acceleration
Dividing both sides of the equation by 0.93 kg/m^3:
acceleration = (0.36 kg/m^3 * g) / 0.93 kg/m^3
acceleration ≈ 0.387 * g
Therefore, the acceleration of the rising balloon is approximately 0.387 times the acceleration due to gravity.