A spherical raindrop of mass 0.0108 g and radius 1.37 mm falls from a cloud that is at a height of 1165 m above the ground. Assume the drag coefficient for the raindrop is 0.60 and the density of the air is 1.3 kg/m3. What is the raindrop's terminal speed?

What would be the raindrop's speed just before landing on the ground if there were no drag force (no air resistance)?

To find the raindrop's terminal speed, we can use the concept of terminal velocity. Terminal velocity is the constant speed reached by an object when the drag force acting upon it is equal to the gravitational force pulling it down.

To calculate the terminal speed, we need to determine the forces acting on the raindrop:

1. Gravitational force (Fg):
The gravitational force is given by Fg = mg, where m is the mass of the raindrop and g is the acceleration due to gravity.

In this case, mass (m) = 0.0108 g = 0.0108 x 10^-3 kg
Acceleration due to gravity (g) ≈ 9.8 m/s^2

2. Drag force (Fd):
The drag force can be calculated using the equation Fd = 0.5 * ρ * Cd * A * v², where ρ is the density of the air, Cd is the drag coefficient, A is the cross-sectional area of the raindrop, and v is the velocity of the raindrop.

Cross-sectional area (A) = π * r², where r is the radius of the raindrop.
In this case, radius (r) = 1.37 mm = 1.37 x 10^-3 m

Now, we can calculate the terminal speed:

1. Calculate the cross-sectional area (A):
A = π * (1.37 x 10^-3)^2

2. Calculate the gravitational force (Fg):
Fg = (0.0108 x 10^-3) * 9.8

3. Set up the equation for terminal velocity:
Fg = 0.5 * ρ * Cd * A * v²

4. Rearrange the equation to solve for v:
v = sqrt((2 * Fg) / (ρ * Cd * A))

Substitute the known values into the equation and calculate the terminal speed.

To find the raindrop's terminal speed, we can use the concept of terminal velocity. Terminal velocity is reached when the drag force on an object (in this case, the raindrop) is equal to the gravitational force pulling it down.

Step 1: Calculate the volume of the raindrop.
The volume of a sphere can be calculated using the formula: V = (4/3) * π * r^3
where V is the volume and r is the radius.
Substituting the given radius of 1.37 mm (or 0.00137 m), we can find:
V = (4/3) * π * (0.00137 m)^3

Step 2: Calculate the density of the raindrop.
Density (ρ) can be calculated by dividing the mass (m) by the volume (V):
ρ = m / V

Step 3: Calculate the gravitational force acting on the raindrop.
The gravitational force (Fg) can be calculated using the formula: Fg = m * g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).

Step 4: Calculate the drag force acting on the raindrop.
The drag force (Fd) can be calculated using the formula: Fd = (1/2) * Cd * ρ * A * v^2
where Cd is the drag coefficient (given as 0.60), A is the cross-sectional area of the raindrop, and v is the speed of the raindrop.

Step 5: Set up the equation for terminal velocity.
At terminal velocity, the drag force Fd is equal to the gravitational force Fg. So we can equate the two equations:
Fd = Fg
(1/2) * Cd * ρ * A * v^2 = m * g

Step 6: Solve for the terminal speed (v).
Rearrange the equation to isolate v:
v^2 = (2 * m * g) / (Cd * ρ * A)
Then take the square root of both sides to find v.

Now, let's calculate the raindrop's terminal speed:

Given:
Mass of the raindrop (m) = 0.0108 g = 0.0108 / 1000 kg
Radius of the raindrop (r) = 1.37 mm = 0.00137 m
Drag coefficient (Cd) = 0.60
Density of air (ρ) = 1.3 kg/m^3
Acceleration due to gravity (g) = 9.8 m/s^2

Step 1: Calculate the volume of the raindrop.
V = (4/3) * π * (0.00137 m)^3

Step 2: Calculate the density of the raindrop.
ρ = m / V

Step 3: Calculate the gravitational force acting on the raindrop.
Fg = m * g

Step 4: Calculate the drag force acting on the raindrop.
Fd = (1/2) * Cd * ρ * A * v^2

Step 5: Solve for the terminal velocity.
(1/2) * Cd * ρ * A * v^2 = m * g

Step 6: Solve for the terminal speed (v).
v = sqrt((2 * m * g) / (Cd * ρ * A))

To find the speed just before landing on the ground with no air resistance, we can use the principle of conservation of energy.

Step 7: Calculate the speed with no air resistance.
Potential energy (PE) = m * g * h
Kinetic energy (KE) = (1/2) * m * v^2, where v is the speed just before landing.
Since there is no air resistance, all the potential energy is converted to kinetic energy.
PE = KE
m * g * h = (1/2) * m * v^2

Step 8: Solve for the speed (v).
v = sqrt((2 * m * g * h) / m)

Now, let's calculate the raindrop's speed just before landing on the ground with no drag force:

Given:
Mass of the raindrop (m) = 0.0108 g = 0.0108 / 1000 kg
Height (h) = 1165 m
Acceleration due to gravity (g) = 9.8 m/s^2

Step 7: Calculate the speed with no air resistance.
Potential energy (PE) = m * g * h
Kinetic energy (KE) = (1/2) * m * v^2

Step 8: Solve for the speed (v).
v = sqrt((2 * m * g * h) / m)