Final answer:
To solve the skydiver's free fall and parachute descent problem, one must set up differential equations accounting for gravity and air resistance for both phases of the fall, integrate to find velocity and distance fallen, and solve for terminal velocity and time in the air once the parachute is opened.
Step-by-step explanation:
In the scenario presented, we will solve a physics problem involving the free fall and parachute descent of a skydiver. Initially, during the free fall, the only forces acting on the skydiver are gravity and air resistance. When the parachute is opened, the air resistance changes, and hence the dynamics of the fall will change as well.
Set up the differential equations
During free fall (before the parachute is opened), the skydiver's acceleration can be described by Newton's second law:
F_{net} = m * a
where F_{net} is the net force, m is the mass of the skydiver, and a is the acceleration. If we consider upward to be positive, then we have:
mg - bv = m * dv/dt
where mg is the weight of the skydiver, b is the drag coefficient, v is the velocity, and dv/dt is the acceleration (change in velocity over time). The gravitational force mg is positive because it acts downward (in the opposite direction of our positive direction) and bv is negative since it is air resistance working opposite to the velocity.
Calculating the needed quantities
- (a) To find the speed of the skydiver when the parachute opens, we need to integrate the differential equation over the 10s of free fall.
- (b) The distance fallen before the parachute opens can be found by integrating the velocity over the 10s of free fall.
- (c) The terminal velocity after the parachute opens can be found by setting the net force equal to zero (mg = bv) and solving for v.
- (d) The time the skydiver is in the air after the parachute opens can be determined by calculating how long it takes for the skydiver to fall the remaining distance at or close to terminal velocity.