During a violent thunderstorm, hail the size of marbles (diameter = 2.10 cm) falls at a speed of 37.0 m/s. There are estimated to be 175.0 hailstones per cubic meter of air. Assume that, as for ice, 1.00 cm³ of hail has a mass of 0.920 g. What is the mass of each hailstone ?
m₀=ρV=ρ•πD³/6
ρ =920 kg/m³, D=0.021 m
To find the mass of each hailstone, we can use the given information about the density of hailstones and their size.
First, let's calculate the volume of each hailstone:
The diameter of the hailstone is 2.10 cm, so the radius would be half of that, which is 1.05 cm (or 0.0105 m).
The volume of a hailstone can be calculated using the formula for the volume of a sphere:
V = (4/3) * π * r³
V = (4/3) * 3.14 * (0.0105)³
V = 1.37 x 10^-5 m³
Now, let's find the mass of each hailstone:
We are given that 1.00 cm³ of hail has a mass of 0.920 grams.
Since the volume of each hailstone is in cubic meters, we need to convert the given mass to kilograms:
0.920 grams = 0.00092 kilograms
Now we can find the mass of each hailstone by multiplying the volume by the mass density:
Mass = Volume * Mass Density
Mass = 1.37 x 10^-5 m³ * 0.00092 kg/m³
Mass = 1.26 x 10^-8 kg
Therefore, the mass of each hailstone is approximately 1.26 x 10^-8 kilograms.
To determine the mass of each hailstone, we can follow these steps:
Step 1: Calculate the volume of each hailstone.
Given that the diameter of each hailstone is 2.10 cm, we can find its volume using the formula for the volume of a sphere, V = (4/3)πr³, where r is the radius of the hailstone.
The radius can be found by dividing the diameter by 2: r = 2.10 cm / 2 = 1.05 cm.
Substituting the radius into the volume formula, V = (4/3)π(1.05 cm)³.
Step 2: Convert the volume into cubic meters.
The volume of the hailstone is currently in cubic centimeters (cm³), but we need to convert it to cubic meters (m³) to match the given density of hailstones per cubic meter.
To convert cm³ to m³, we divide by 1,000,000 (since there are 1,000,000 cm³ in 1 m³).
Step 3: Calculate the mass of each hailstone.
Given that 1.00 cm³ of hail has a mass of 0.920 g, we can determine the mass of each hailstone by multiplying the volume in cubic meters by the density in g/cm³.
The density is given as 0.920 g/cm³.
Therefore, the mass of each hailstone is the volume in m³ multiplied by the density: Mass = V (m³) * 0.920 g/cm³.
Let's perform the calculations to find the mass of each hailstone:
V = (4/3)π(1.05 cm)³ = (4/3)π(1.05 cm)³ ≈ 4.886 cm³
Converting to m³: V (m³) = 4.886 cm³ ÷ 1,000,000 = 0.000004886 m³
Mass = V (m³) * 0.92 g/cm³ = 0.000004886 m³ * 0.920 g/cm³ ≈ 0.000004489 g
Therefore, each hailstone has a mass of approximately 0.000004489 grams.