(b) If the skydiver's daughter, whose mass is 45 kg, is falling through the air and has the same D (0.25 kg/m) as her father, what is the daughter's terminal speed?

Verified step by step guidance

1. Identify the formula for terminal speed in the context of a falling object with air resistance. The formula is: \( v_t = \sqrt{\frac{2mg}{\rho C_d A}} \), where \( v_t \) is the terminal speed, \( m \) is the mass of the object, \( g \) is the acceleration due to gravity (approximately 9.81 m/s^2), \( \rho \) is the density of air (approximately 1.225 kg/m^3), \( C_d \) is the drag coefficient, and \( A \) is the cross-sectional area of the object.
2. Given that the drag coefficient per unit area (D) is provided as 0.25 kg/m, and knowing that \( D = \rho C_d A \), solve for \( C_d A \) using the equation: \( C_d A = \frac{D}{\rho} \).
3. Substitute the value of \( C_d A \) from step 2 into the terminal speed formula.
4. Plug in the values for \( m \) (45 kg) and \( g \) (9.81 m/s^2) into the terminal speed formula.
5. Calculate the square root of the expression obtained in step 4 to find the terminal speed, \( v_t \), of the skydiver's daughter.

Key Concepts

Terminal Velocity

Terminal velocity is the constant speed an object reaches when the force of gravity pulling it down is balanced by the drag force acting against it. At this point, the net force on the object is zero, resulting in no further acceleration. For a skydiver, terminal velocity depends on factors such as mass, cross-sectional area, and drag coefficient.

Drag Force

The drag force is the resistance experienced by an object moving through a fluid, such as air. It is influenced by the object's speed, shape, and the fluid's density. The drag force can be calculated using the equation F_d = 0.5 * C_d * ρ * A * v^2, where C_d is the drag coefficient, ρ is the fluid density, A is the cross-sectional area, and v is the velocity.

Mass and Weight

Mass is a measure of the amount of matter in an object, typically measured in kilograms. Weight, on the other hand, is the force exerted by gravity on that mass, calculated as W = m * g, where g is the acceleration due to gravity (approximately 9.81 m/s² on Earth). In the context of falling objects, a greater mass results in a higher weight, which influences the terminal velocity.