Question

a skydiver is falling through the air attached to a parachute. the mass of the skydiver - parachute system is m. the upward force of air resistance exerted on the system is given by the equation ( f_{\text{air}} = av^{2} ), where a is a constant that depends on the shape of the parachute and v is the speed of the system. after falling a short time, the system reaches a constant speed ( v_{t} ). which of the following is a correct expression for ( v_{t} )?
a 0
b ( sqrt{\frac{g}{a}} )
c ( sqrt{\frac{mg}{a}} )
d ( sqrt{\frac{ag}{m}} )

Explanation:

Step1: Analyze the forces when speed is constant

When the system reaches a constant speed \(v_T\), the net force on it is zero. According to Newton's second law \(F_{net}=ma\), and since \(a = 0\) (constant - speed motion), the upward force \(F_{air}\) equals the downward force (weight \(mg\)). So, \(F_{air}=mg\).

Step2: Substitute \(F_{air}\) into its formula

We know that \(F_{air}=Av_T^{2}\). Substituting \(F_{air}=mg\) into this equation gives \(Av_T^{2}=mg\).

Step3: Solve for \(v_T\)

To find \(v_T\), we first isolate \(v_T^{2}\): \(v_T^{2}=\frac{mg}{A}\). Then take the square - root of both sides. Since speed \(v_T>0\), \(v_T=\sqrt{\frac{mg}{A}}\).

Answer:

C. \(\sqrt{\frac{mg}{A}}\)