Velocity of falling body Refer to Exercise 95, which gives the position function for a falling body. Use m = 75 kg and k = 0.2.


a. Confirm that the BASE jumper’s velocity t seconds after jumping is v(t) = d'(t) = √(mg/k) tanh (√(kg/m) t).

Verified step by step guidance

1. Recall that the position function for the falling body is given by \(d(t)\), and the velocity function \(v(t)\) is the derivative of the position function with respect to time, i.e., \(v(t) = d'(t)\).
2. Identify the constants given: mass \(m = 75\) kg and drag coefficient \(k = 0.2\). Also, gravitational acceleration \(g\) is typically \$9.8 \ \text{m/s}^2$ unless otherwise specified.
3. Express the velocity function in terms of \(m\), \(k\), and \(g\). The problem states that \(v(t) = \sqrt{\frac{mg}{k}} \tanh \left( \sqrt{\frac{kg}{m}} t \right)\), so we need to confirm this by differentiating the position function \(d(t)\).
4. Use the chain rule to differentiate \(d(t)\), which likely involves hyperbolic functions due to the presence of \(\tanh\) in \(v(t)\). The derivative of \(\tanh(x)\) is \(\text{sech}^2(x)\), and the derivative inside the argument must be accounted for.
5. After differentiating, simplify the expression to show that it matches the given velocity formula \(v(t) = \sqrt{\frac{mg}{k}} \tanh \left( \sqrt{\frac{kg}{m}} t \right)\), confirming the velocity function.

Key Concepts

Position and Velocity Functions

The position function d(t) describes the location of a falling body at time t, while the velocity function v(t) is its derivative d'(t), representing the rate of change of position. Understanding how to differentiate position functions is essential to find velocity.

Hyperbolic Functions and Their Derivatives

Hyperbolic functions like tanh(x) often appear in solutions to differential equations involving drag forces. Knowing the properties and derivatives of tanh(x) helps verify velocity expressions derived from position functions.

Modeling Drag Force in Falling Bodies

The parameters m (mass) and k (drag coefficient) model the effect of air resistance on a falling body. The velocity formula involving √(mg/k) and tanh(√(kg/m) t) arises from solving the motion equation with drag proportional to velocity.