Hail forms high in the atmosphere and can be accelerated to a high speed before it reaches the ground. Estimate the terminal speed of a spherical hailstone that has a diameter of 2.8 cm. Hint: The mass of a piece of hail that has a volume of 1 cm3 is about 1 g.
You'll need to use:
en.wikipedia(dot)org/wiki/Drag_(physics)
to calculate at what speed the
Fz + Fdrag = 0
To estimate the terminal speed of a spherical hailstone, we need to consider the forces acting on it. The two main forces to consider are the gravitational force pulling the hailstone downward and the air resistance force pushing against the hailstone as it falls.
We can start by calculating the mass of the hailstone using its volume. The given hint tells us that the mass of a hailstone with a volume of 1 cm^3 is about 1 gram. Since the hailstone is spherical, we can use the formula for the volume of a sphere:
V = (4/3) * pi * r^3,
where V is the volume and r is the radius.
Given that the diameter of the hailstone is 2.8 cm, we can find the radius by dividing the diameter by 2:
r = 2.8 cm / 2 = 1.4 cm = 0.014 m.
Substituting this value into the formula for volume:
V = (4/3) * pi * (0.014 m)^3 ≈ 9.523 x 10^-6 m^3.
Now, we can calculate the mass of the hailstone using the given mass-to-volume ratio:
Mass = V * mass-to-volume ratio = 9.523 x 10^-6 m^3 * 1 g/cm^3 ≈ 9.523 x 10^-6 kg.
Next, we need to consider the forces acting on the hailstone. The gravitational force, F_gravity, is given by:
F_gravity = m * g,
where m is the mass of the hailstone and g is the acceleration due to gravity (approximately 9.8 m/s^2).
F_gravity = 9.523 x 10^-6 kg * 9.8 m/s^2 ≈ 9.3364 x 10^-5 N.
The air resistance force, F_air, is proportional to the speed of the hailstone and can be expressed as:
F_air = 0.5 * C_d * A * ρ * v^2,
where C_d is the drag coefficient (which we'll assume to be 0.47 for a sphere), A is the cross-sectional area of the hailstone perpendicular to its motion, ρ is the air density (approximately 1.2 kg/m^3), and v is the speed of the hailstone.
The cross-sectional area of a sphere is given by:
A = pi * r^2.
Plugging in the radius and other values:
A = pi * (0.014 m)^2 ≈ 6.154 x 10^-4 m^2.
Finally, we can estimate the terminal speed by assuming that the gravitational force and the air resistance force are equal:
F_gravity = F_air.
Solving for v, the terminal speed:
v = sqrt((2 * m * g) / (C_d * A * ρ)).
Substituting in the known values:
v = sqrt((2 * 9.523 x 10^-6 kg * 9.8 m/s^2) / (0.47 * 6.154 x 10^-4 m^2 * 1.2 kg/m^3)) ≈ 15.3 m/s.
Therefore, the estimated terminal speed of the spherical hailstone with a diameter of 2.8 cm is approximately 15.3 m/s.