A child loses his balloon, which rises slowly into the sky. If the balloon is 17 cm in diameter when the child loses it, what is its diameter at the following altitudes? Assume that the balloon is very flexible and so surface tension can be neglected.
(a) 1100 m
(b) 1900 m
(c) 5000 m
To find the diameter of the balloon at different altitudes, we need to consider the change in atmospheric pressure with altitude. As the altitude increases, the atmospheric pressure decreases, causing the balloon to expand.
To calculate the diameter of the balloon at a specific altitude, we can use the concept of Boyle's law, which states that the volume of a gas is inversely proportional to the pressure exerted on it, as long as the temperature remains constant.
Let's assume the initial pressure and diameter of the balloon are P₁ and D₁, respectively, when the child loses it. At a higher altitude, the pressure decreases to P₂, and we need to find the new diameter, D₂.
We can use the equation for Boyle's law as follows:
(P₁)(V₁) = (P₂)(V₂)
Since the volume of a balloon can be approximated as proportional to the cube of its diameter, we can rewrite the equation as:
(P₁)(D₁³) = (P₂)(D₂³)
Now, let's plug in the given information and solve for D₂ for each altitude:
(a) At 1100 m:
The atmospheric pressure at this altitude is significantly lower than at ground level. We can assume the pressure difference is negligible when compared to the initial pressure (P₁). Therefore, P₂ ≈ 0.
(0)(D₁³) = (P₂)(D₂³)
0 = D₂³
Since the cube of D₂ is zero, D₂ must be zero as well. Thus, the diameter of the balloon at 1100 m is zero.
(b) At 1900 m:
Similarly to the previous case, the pressure difference can be considered negligible compared to the initial pressure (P₁). Therefore, P₂ ≈ 0.
(0)(D₁³) = (P₂)(D₂³)
0 = D₂³
Again, the cube of D₂ is zero, so the diameter of the balloon at 1900 m is also zero.
(c) At 5000 m:
At this altitude, the atmospheric pressure is significantly lower than at ground level, but the pressure difference is not negligible.
(P₁)(D₁³) = (P₂)(D₂³)
To find P₂, we can use the barometric formula, which describes how atmospheric pressure decreases with altitude. The formula is given as:
P = P₀ * (1 - (L * h / T₀))^((g * M) / (R * L))
Where:
P is the pressure at the given altitude,
P₀ is the pressure at sea level,
L is the temperature lapse rate (approximated as 0.0065 K/m),
h is the altitude,
T₀ is the temperature at sea level (approximated as 288.15 K),
g is the acceleration due to gravity (approximated as 9.81 m/s²),
M is the molar mass of Earth's air (approximated as 0.02896 kg/mol), and
R is the universal gas constant (approximated as 8.314 J/(mol*K)).
Using the barometric formula, we can calculate P₂. Then we can substitute the values into the original equation:
(P₁)(D₁³) = (P₂)(D₂³)
Calculate the diameter of the balloon at 5000 m using the provided equations and the given values.