1. **Problem statement:** A skydiver jumps from 4,000 meters above the ground. We need to calculate the time to fall this distance under gravity without air resistance, the velocity just before parachute deployment, and discuss effects of air resistance and parachute design. 2. **Formula for free fall time:** The distance fallen under constant acceleration $g$ is given by $$ h = \frac{1}{2} g t^2 $$ where $h=4000$ m and $g=9.8$ m/s$^2$. 3. **Calculate time $t$:** $$ t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 \times 4000}{9.8}} $$ $$ t = \sqrt{\frac{8000}{9.8}} = \sqrt{816.33} \approx 28.58 \text{ seconds} $$ 4. **Calculate velocity just before parachute deployment:** Velocity under free fall is $$ v = g t = 9.8 \times 28.58 = 280.08 \text{ m/s} $$ 5. **Explain air resistance and terminal velocity:** In reality, air resistance opposes motion and increases with velocity, causing the skydiver to reach a terminal velocity where acceleration stops and velocity becomes constant, much less than 280 m/s. 6. **Parachute design and deployment timing:** Parachutes increase air resistance drastically, reducing velocity to safe landing speeds. Deploying too late or too early can be dangerous; timing ensures the skydiver slows down sufficiently before landing. **Final answers:** - Time to fall 4000 m without air resistance: $\approx 28.58$ seconds - Velocity just before parachute deployment (idealized): $\approx 280.08$ m/s