Terminal velocity Refer to Exercises 95 and 96.


d. How tall must a cliff be so that the BASE jumper (m = 75 kg and k = 0.2) reaches 95% of terminal velocity? Assume the jumper needs at least 300 m at the end of free fall to deploy the chute and land safely.

Verified step by step guidance

1. Recall the velocity function for an object falling with air resistance proportional to velocity: \(v(t) = v_{\text{terminal}} (1 - e^{-\frac{k}{m} t})\), where \(v_{\text{terminal}} = \frac{mg}{k}\).
2. Calculate the terminal velocity \(v_{\text{terminal}}\) using the given mass \(m = 75\) kg, gravitational acceleration \(g = 9.8\) m/s², and drag coefficient \(k = 0.2\): \(v_{\text{terminal}} = \frac{75 \times 9.8}{0.2}\).
3. Set the velocity to 95% of terminal velocity: \(v(t) = 0.95 \times v_{\text{terminal}}\), and solve for time \(t\) using the velocity formula: \$0.95 = 1 - e^{-\frac{k}{m} t}$.
4. Rearrange to isolate the exponential term: \(e^{-\frac{k}{m} t} = 1 - 0.95 = 0.05\), then take the natural logarithm to solve for \(t\): \(t = -\frac{m}{k} \ln(0.05)\).
5. Find the distance fallen during time \(t\) by integrating the velocity function or using the position formula: \(s(t) = v_{\text{terminal}} \left(t + \frac{m}{k} e^{-\frac{k}{m} t}\right)\). Add the 300 m needed for chute deployment to find the total cliff height.

Key Concepts

Terminal Velocity

Terminal velocity is the constant speed an object reaches when the force of gravity is balanced by the drag force from air resistance. At this point, acceleration stops, and the object falls at a steady speed. It depends on factors like mass, drag coefficient, and air density.

Differential Equations in Motion with Air Resistance

The motion of a falling object with air resistance is modeled by a differential equation balancing gravitational force and drag force. Solving this equation gives velocity as a function of time or distance, which is essential to determine how long or how far it takes to reach a certain fraction of terminal velocity.

Free Fall Distance and Safety Constraints

In problems involving BASE jumping, the total height must account for the distance needed to reach a target velocity and the safety margin for parachute deployment. Understanding how to partition the total height into acceleration and deceleration zones ensures the jumper can safely land.