Falling body When an object falling from rest encounters air resistance proportional to the square of its velocity, the distance it falls (in meters) after t seconds is given by d(t) = (m/k) ln (cosh (√(kg/m) t)), where m is the mass of the object in kilograms, g = 9.8 m/s² is the acceleration due to gravity, and k is a physical constant.


a. A BASE jumper (m = 75 kg) leaps from a tall cliff and performs a ten-second delay (she free-falls for 10 s and then opens her chute). How far does she fall in 10 s? Assume k = 0.2.

Verified step by step guidance

1. Identify the given function for the distance fallen: \(d(t) = \frac{m}{k} \ln \left( \cosh \left( \sqrt{\frac{kg}{m}} \, t \right) \right)\), where \(m\) is mass, \(k\) is a constant, \(g = 9.8\) m/s², and \(t\) is time in seconds.
2. Substitute the known values into the formula: \(m = 75\), \(k = 0.2\), \(g = 9.8\), and \(t = 10\) seconds.
3. Calculate the term inside the square root: \(\sqrt{\frac{kg}{m}} = \sqrt{\frac{0.2 \times 9.8}{75}}\).
4. Evaluate the argument of the hyperbolic cosine function: \(\sqrt{\frac{kg}{m}} \times t = \left( \text{value from previous step} \right) \times 10\).
5. Compute the distance fallen by plugging the value into the formula: \(d(10) = \frac{75}{0.2} \times \ln \left( \cosh \left( \text{value from previous step} \right) \right)\).

Key Concepts

Modeling Air Resistance in Free Fall

Air resistance affects falling objects by opposing motion, often modeled as proportional to velocity squared for high speeds. This nonlinear drag force changes the acceleration and velocity over time, making the motion differ from simple free fall. Understanding this helps interpret the given distance formula involving hyperbolic functions.

Hyperbolic Functions and Their Properties

Hyperbolic functions like cosh(x) and sinh(x) arise in solutions to differential equations involving quadratic velocity terms. The function cosh(x) = (e^x + e^{-x})/2 grows exponentially and appears in the distance formula, reflecting the balance between gravity and air resistance in the falling motion.

Applying Given Formulas with Physical Constants

To find the distance fallen, substitute the known values (mass m, gravity g, constant k, and time t) into the formula d(t) = (m/k) ln(cosh(√(kg/m) t)). This requires careful calculation of the square root term and the natural logarithm, ensuring units are consistent and the physical meaning is preserved.