A 570-g squirrel with a surface area of 900 cm2 falls from a 4.0-m tree to the ground. Estimate its terminal velocity. (Use the drag coefficient for a horizontal skydiver. Assume that the cross-sectional area of the squirrel can be approximated as a rectangle of width 11.3 cm and length 22.6 cm. Note, the squirrel may not reach terminal velocity by the time it hits the ground. Give the squirrel's terminal velocity, not it's velocity as it hits the ground.)
To estimate the squirrel's terminal velocity, we can use the equation for drag force:
Fd = (1/2) * ρ * v^2 * A * Cd
Where:
- Fd is the drag force
- ρ (rho) is the density of the fluid the object is moving through (air, in this case)
- v is the velocity of the object
- A is the cross-sectional area of the object perpendicular to the direction of motion
- Cd is the drag coefficient
In this case, we assume the squirrel is falling vertically through the air, so we can ignore the weight of the squirrel since it is balanced by the drag force. Terminal velocity occurs when the drag force equals the gravitational force acting on the squirrel.
Fg = m * g
Where:
- Fg is the gravitational force
- m is the mass of the squirrel
- g is the acceleration due to gravity
To find the terminal velocity, we need to solve for v in the equation Fd = Fg. Rearranging the equation:
(1/2) * ρ * v^2 * A * Cd = m * g
Simplifying:
v^2 = (2 * m * g) / (ρ * A * Cd)
Taking the square root of both sides gives us:
v = √[(2 * m * g) / (ρ * A * Cd)]
Now let's calculate the terminal velocity step by step:
1. First, convert the mass of the squirrel from grams to kilograms:
Mass (m) = 570 g = 0.57 kg
2. Calculate the cross-sectional area (A) of the squirrel:
A = width * length
A = 11.3 cm * 22.6 cm = 254.38 cm^2
3. Convert the cross-sectional area to square meters:
A = 254.38 cm^2 = 254.38 cm^2 * (1 m^2 / 10,000 cm^2) = 0.025438 m^2
4. Determine the values of density (ρ) and drag coefficient (Cd):
For a horizontal skydiver, the drag coefficient (Cd) is approximately 0.8.
The density of air (ρ) is around 1.225 kg/m^3.
5. Substitute the values into the equation for terminal velocity:
v = √[(2 * m * g) / (ρ * A * Cd)]
v = √[(2 * 0.57 kg * 9.8 m/s^2) / (1.225 kg/m^3 * 0.025438 m^2 * 0.8)]
Now you can calculate the terminal velocity of the squirrel by plugging in the values and solving for v.