A child loses his balloon, which rises slowly into the sky.
a) If the balloon is 21.5 cm in diameter when the child loses it, what is its diameter at an altitude of 1150 m? Assume that the balloon is very flexible and so surface tension can be neglected.
b) And what is the diameter of the same balloon at an altitude of 2150 m?
c) And what is the diameter of the same balloon at an altitude of 5150 m?
To find the diameter of the balloon at different altitudes, we need to take into account the change in pressure as the balloon rises. The pressure decreases with increasing altitude. We can use the ideal gas law and combine it with the hydrostatic pressure equation to solve for the diameter at different altitudes.
a) To find the diameter of the balloon at an altitude of 1150 m, we first need to calculate the pressure at that altitude. The hydrostatic pressure equation is given by:
P = P₀ + ρgh
Where P is the pressure at the given altitude, P₀ is the initial pressure at ground level, ρ is the density of air, g is the acceleration due to gravity, and h is the height or altitude.
We can assume that P₀ is equal to atmospheric pressure at ground level, which is approximately 101325 Pascal (Pa). The density of air, ρ, is approximately 1.225 kg/m³, and the acceleration due to gravity, g, is 9.81 m/s².
Plugging in the values, we get:
P = 101325 Pa + (1.225 kg/m³ * 9.81 m/s² * 1150 m)
Now, we can use the ideal gas law to relate the pressure, volume, and temperature of the gas inside the balloon. Assuming the balloon is filled with air, we can use the ideal gas law equation:
PV = nRT
Where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin.
Since we are interested in the diameter of the balloon, we can assume that the volume of the balloon remains constant as it rises. Therefore, we can rewrite the ideal gas law equation as:
P₁ * V₁ = P₂ * V₂
Where P₁ is the pressure at ground level, V₁ is the initial volume (which can be calculated from the initial diameter), P₂ is the pressure at the given altitude, and V₂ is the volume at the given altitude.
If we solve the equation for V₂, we get:
V₂ = (P₁ * V₁) / P₂
Now we can calculate the new diameter at the given altitude using the formula for the volume of a sphere:
V = (4/3) * π * (d/2)³
Where V is the volume and d is the diameter.
Rearranging the formula to solve for the diameter, we get:
d = (3V / (4π))^(1/3)
Substituting V₂ into the formula, we can calculate the new diameter.
To answer these questions, we can use the ideal gas law, which states that the pressure and volume of a gas are inversely proportional when temperature is held constant. In this case, the balloon is filled with air, so we can assume the volume inside the balloon remains constant as it rises.
a) To find the new diameter of the balloon at an altitude of 1150 m, we can use the concept of Boyle's Law. Boyle's Law states that the pressure and volume of a gas are inversely proportional. As the balloon rises, the pressure decreases. We can use the equation P1V1 = P2V2, where P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume.
Let's assume the air pressure at sea level is 1 atmosphere (~101325 Pascals) and the air pressure at an altitude of 1150 m is 0.9 atmosphere (~91297.5 Pascals). Since the volume remains constant, we can rewrite Boyle's Law as P1/P2 = V2/V1.
Let's represent the initial diameter of the balloon as D1 and the final diameter at an altitude of 1150 m as D2. Since the diameter is directly proportional to the volume, we can rewrite the equation as P1/P2 = (D2/D1)^2, where the ratio of diameters is squared.
Plugging in the values, we get (1/0.9) = (D2/21.5)^2. Solving for D2, we find:
(D2/21.5)^2 = 1.1111
D2/21.5 = sqrt(1.1111)
D2 = sqrt(1.1111) * 21.5
Calculating this, we find that the diameter of the balloon at an altitude of 1150 m is approximately 22.13 cm.
b) Following the same approach, we can calculate the diameter of the balloon at an altitude of 2150 m. Assuming the air pressure at this altitude to be 0.8 atmospheres (~81060 Pascals), we can use the equation P1/P2 = (D2/D1)^2 again.
(1/0.8) = (D2/21.5)^2
(D2/21.5)^2 = 1.25
D2 = √(1.25) * 21.5
Calculating this, we find that the diameter of the balloon at an altitude of 2150 m is approximately 23.24 cm.
c) Applying the same logic, we can calculate the diameter of the balloon at an altitude of 5150 m. Assuming the air pressure at this altitude to be 0.6 atmospheres (~60795 Pascals), we can use the equation:
(1/0.6) = (D2/21.5)^2
(D2/21.5)^2 = 1.6667
D2 = √(1.6667) * 21.5
Calculating this, we find that the diameter of the balloon at an altitude of 5150 m is approximately 24.93 cm.