Hail of a diameter of 1 cm fall at a constant speed of 25 m/s. There is estimated to be 120 hailstones per cubic meter of air. Mass of each hailstone is .482g. Assuming the hail does not bounce, find the average force on a flat roof measuring 10m by 20m, due to the impact of the hail
To find the average force on the roof due to the impact of hail, we need to calculate the total force exerted by all the hailstones that hit the roof.
First, let's find the volume of hailstones that will hit the roof. We can do this by considering the number of hailstones per cubic meter of air. Since there are 120 hailstones per cubic meter of air, the volume of hailstones per cubic meter of air is 120 times the volume of each hailstone.
The volume of a hailstone can be calculated using its diameter. The formula for the volume of a sphere is V = (4/3)πr^3, where V is the volume and r is the radius. In this case, the radius (half the diameter) of the hailstone is 0.5 cm or 0.005 m.
So, the volume of each hailstone is (4/3)π(0.005)^3 = 0.000052 m^3.
Now, let's calculate the total volume of hailstones hitting the roof. The area of the roof is 10m * 20m = 200m^2. Since we want the volume per cubic meter of air, we divide the area by the thickness of the hailstorm, which is 1m (assuming the hailstorm is 1m thick).
Therefore, the total volume of hailstones hitting the roof is 200m^2 * 1m = 200m^3.
Next, let's find the total mass of the hailstones hitting the roof. We know that the mass of each hailstone is 0.482g, which is 0.482/1000 kg = 0.000482 kg.
The total mass of hailstones hitting the roof is the product of the total volume and the mass of each hailstone: 200m^3 * 0.000482 kg = 0.0964 kg.
Now, let's calculate the average force on the roof. The average force is equal to the change in momentum divided by the time it takes to stop.
The change in momentum is given by the momentum of the hailstones impacting the roof, which is the mass of the hailstones multiplied by their velocity. In this case, the velocity is 25 m/s.
The change in momentum is 0.0964 kg * 25 m/s = 2.41 kg·m/s.
Since the hail does not bounce, we can assume it comes to rest within a very short time, so we can approximate the time it takes to stop as 0.
Therefore, the average force on the roof due to the hail impact is 2.41 kg·m/s / 0s = ∞ N (Infinity).
It is important to note that this result means that the impact of the hailstones on the roof is an extremely large force that cannot be accurately determined without more information, such as the actual time it takes for the hailstones to come to rest.
To find the average force on the flat roof due to the impact of the hail, we need to calculate the total force exerted by all the hailstones on the roof and then divide it by the area of the roof.
1. Calculate the volume of each hailstone:
Volume = (4/3) * pi * (radius)^3
Given that the diameter is 1 cm, the radius is 0.5 cm = 0.005 m.
Volume = (4/3) * pi * (0.005)^3 = 5.24 * 10^-8 m^3
2. Calculate the mass of each hailstone:
Given that the mass is 0.482 g, convert it to kg:
Mass = 0.482 g = 0.482 * 10^-3 kg = 4.82 * 10^-4 kg
3. Calculate the total number of hailstones that would fall on the roof:
Given that there are 120 hailstones per cubic meter of air, and the roof has an area of 10m * 20m = 200 m^2.
Total number of hailstones = 120 * 200 = 24000
4. Calculate the total force exerted by all the hailstones on the roof:
Total force = total number of hailstones * mass of each hailstone * acceleration due to gravity
Acceleration due to gravity = 9.8 m/s^2
Total force = 24000 * (4.82 * 10^-4) * 9.8 = 1.129 N
5. Calculate the average force on the roof:
Average force = Total force / Area of the roof
Area of the roof = 10m * 20m = 200 m^2
Average force = 1.129 N / 200 = 0.005645 N
Therefore, the average force on the flat roof due to the impact of the hail is approximately 0.005645 Newtons.