Estimate the terminal speed of a wooden sphere (density 0.770 g/cm3) falling through air, if its radius is 9.00 cm and its drag coefficient is 0.500. (The density of air is 1.20 kg/m3.)
(b) From what height would a freely falling object reach this speed in the absence of air resistance?
To estimate the terminal speed of the wooden sphere falling through air, we can use the drag force equation:
F_drag = 0.5 *ρ * A * C_d * v^2
where:
F_drag is the drag force,
ρ is the density of air (1.20 kg/m^3),
A is the cross-sectional area of the sphere,
C_d is the drag coefficient (0.500), and
v is the velocity of the sphere.
The weight of the sphere can be determined using:
F_weight = m * g
where:
m is the mass of the sphere, and
g is the acceleration due to gravity (9.8 m/s^2).
At terminal velocity, the drag force is equal to the weight of the sphere. So we have:
F_drag = F_weight
By equating the drag force equation with the weight equation, we can solve for the terminal velocity v.
First, let's determine the mass of the wooden sphere:
Density = mass / volume
0.770 g/cm^3 = mass / (4/3 * π * r^3)
mass = density * (4/3 * π * r^3)
(Here, r is the radius of the sphere, which is given as 9.00 cm.)
Next, let's calculate the cross-sectional area of the sphere:
A = π * r^2
Now, let's substitute the equations into the drag force equation and solve for the terminal velocity:
F_drag = 0.5 * ρ * A * C_d * v^2
m * g = 0.5 * ρ * A * C_d * v^2
(mass * g) / (0.5 * ρ * A * C_d) = v^2
√[(mass * g) / (0.5 * ρ * A * C_d)] = v
Substituting all the known values:
mass = 0.770 g/cm^3 * (4/3 * π * (9.00 cm)^3)
A = π * (9.00 cm)^2
g = 9.8 m/s^2
ρ = 1.20 kg/m^3
C_d = 0.500
We can now calculate the terminal velocity using the given values.
To estimate the terminal speed of the wooden sphere falling through air, we need to calculate the force of gravity and the drag force acting on the sphere as it falls.
Step 1: Calculate the force of gravity acting on the sphere.
The force of gravity can be calculated using the formula:
F_gravity = m * g
where m is the mass of the sphere and g is the acceleration due to gravity.
To find the mass of the sphere, we can use the formula:
m = density * volume
The volume of a sphere can be calculated using the formula:
V = (4/3) * π * r^3
Given:
Density of the sphere (ρ) = 0.770 g/cm^3
Radius of the sphere (r) = 9.00 cm = 0.09 m
Density of air (ρ_air) = 1.20 kg/m^3
Acceleration due to gravity (g) = 9.8 m/s^2
Step 2: Calculate the volume and mass of the sphere.
V = (4/3) * π * r^3
V = (4/3) * π * (0.09)^3
V ≈ 0.305 m^3
m = ρ * V
m = 0.770 g/cm^3 * 0.305 m^3
m ≈ 0.235 kg
Step 3: Calculate the force of gravity and drag force.
F_gravity = m * g
F_gravity = 0.235 kg * 9.8 m/s^2
F_gravity ≈ 2.299 N
The drag force can be calculated using the formula:
F_drag = (1/2) * ρ_air * v^2 * A * C_d
where v is the velocity of the sphere, A is the cross-sectional area of the sphere, and C_d is the drag coefficient.
The terminal velocity is reached when the forces are balanced, so the drag force equals the force of gravity:
F_drag = F_gravity
Step 4: Calculate the terminal velocity.
(1/2) * ρ_air * v^2 * A * C_d = m * g
To calculate the cross-sectional area (A) of the sphere, we use the formula:
A = π * r^2
A = π * (0.09)^2
A ≈ 0.025 m^2
Now we can calculate the terminal velocity:
(1/2) * 1.20 kg/m^3 * v^2 * 0.025 m^2 * 0.500 = 0.235 kg * 9.8 m/s^2
0.015 * v^2 ≈ 2.299
v^2 ≈ 2.299 / 0.015
v^2 ≈ 153.3
v ≈ √(153.3)
v ≈ 12.4 m/s
So, the estimated terminal speed of the wooden sphere falling through air is approximately 12.4 m/s.
Now, let's calculate the height from which the object would reach this speed in the absence of air resistance.
In the absence of air resistance, the only force acting on the object is its weight. Therefore, the distance it falls is determined by the gravitational potential energy.
The potential energy (PE) can be calculated using the formula:
PE = m * g * h
We can rearrange this formula to solve for the height (h):
h = PE / (m * g)
PE = 1/2 * m * v^2
PE = 1/2 * 0.235 kg * (12.4 m/s)^2
PE ≈ 18.34 J
h = 18.34 J / (0.235 kg * 9.8 m/s^2)
h ≈ 7.98 m
Therefore, a freely falling object would reach the terminal speed of 12.4 m/s from a height of approximately 7.98 meters in the absence of air resistance.