Question

a skydiver is dropped out of an airplane at an altitude of 10000 feet. he reaches a terminal velocity 40 seconds later. consider four positions during his fall. a: initial state (t = 0 seconds) b: 10 seconds after drop c: 30 seconds after drop d: 40 seconds after drop toggle through the set of vector diagrams at the right to identify the relative magnitude of the air resistance vector for each of these four positions. (consider vertical motion only.)

Explanation:

Step1: Recall Terminal Velocity Concept

Terminal velocity is reached when the net force on the skydiver is zero, meaning air resistance (\(F_{air}\)) equals the force of gravity (\(F_g\)). Before terminal velocity, the skydiver is accelerating downward, so \(F_g > F_{air}\). At \(t = 0\) (State A), the skydiver starts falling, so initial velocity is 0, and air resistance is 0 (since air resistance depends on velocity, \(F_{air} \propto v^2\) or \(v\) for different models, but at \(v = 0\), \(F_{air} = 0\)).

Step2: Analyze State B (10s)

After 10 seconds, the skydiver is moving downward, so air resistance is non - zero, but since the skydiver is still accelerating (not at terminal velocity yet, terminal velocity at 40s), \(F_g>F_{air}\).

Step3: Analyze State C (30s)

At 30 seconds, the skydiver is still accelerating (terminal velocity at 40s), so \(F_g>F_{air}\). As time increases towards 40s, the velocity increases, so air resistance increases (since \(F_{air}\) depends on velocity). So at 30s, \(F_{air}\) is larger than at 10s, but still less than \(F_g\).

Step4: Analyze State D (40s)

At 40 seconds, terminal velocity is reached. So the net force is zero, which means \(F_{air}=F_g\).

Answer:

• State A (\(t = 0\)): Air resistance vector magnitude is \(0\) (since \(v = 0\), \(F_{air}=0\)).
• State B (10s): Air resistance vector magnitude is less than \(F_g\), and less than at State C and State D.
• State C (30s): Air resistance vector magnitude is less than \(F_g\), greater than at State B, and less than at State D.
• State D (40s): Air resistance vector magnitude is equal to \(F_g\) (since terminal velocity is reached, \(F_{air}=F_g\)).