Question

Sky Diving A sky diver jumps from a reasonable height above the ground. The air resistance she experiences is proportional to her velocity, and the constant of proportionality is 0.2, It can be shown that the downward velocity of the sky diver at time t is given by

$v\left(t\right)=180\left(1-e^{-0.2t}\right)$

where f is measured in seconds (s) and v(t) is measured in feet per second (ft/s).

(a) Find the initial velocity of the sky diver.

(b) Find the velocity after 5 s and after 10 s.

(c) Draw a graph of the velocity function v(t).

(d) The maximum velocity of a falling object with wind resistance is called its terminal velocity. From the graph in part (c) find the terminal velocity of this sky diver.

Short answer

a. The initial velocity is 0

b. The velocity after 5s and 10s are 113.78 ft/s and 155.64 ft/s respectively.

c. The required graph is:

d. The terminal velocity is 180 ft/s

Step-by-step solution

Part a. Step 1. Given.

Given that the downward velocity of the sky diver at time t is given by:

$v\left(t\right)=180\left(1-e^{-0.2t}\right)$

Part a. Step 2. To determine.

We have to find the initial velocity.

Part a. Step 3. Calculation.

For initial time $t=0$.

So, for$t=0$ the velocity becomes:

$$\begin{gathered} v\left(0\right)=180\left(1-e^{-0.2\left(0\right)}\right) \\ =180\left(1-e^0\right) \\ =180\left(1-1\right) \\ =180\left(0\right) \\ =0 \end{gathered}$$

Hence, the initial velocity is 0.

Part b. Step 1. Given.

Given that the downward velocity of the sky diver at time t is given by:

$v\left(t\right)=180\left(1-e^{-0.2t}\right)$

Part b. Step 2. To determine.

We have to find the velocity after 5s and 10s.

Part b. Step 3. Calculation.

For 5s and 10s we plug$t=5,10$ respectively.

So, for$t=5$ the velocity becomes:

$$\begin{gathered} v\left(5\right)=180\left(1-e^{-0.2\left(5\right)}\right) \\ =113.78 \end{gathered}$$

So, for$t=10$ the velocity becomes:

$$\begin{gathered} v\left(10\right)=180\left(1-e^{-0.2\left(10\right)}\right) \\ =155.64 \end{gathered}$$

Hence, the velocity after 5s and 10s are 113.78 ft/s and 155.64 ft/s respectively.

Part c. Step 1. Given.

Given that the downward velocity of the sky diver at time t is given by:

$v\left(t\right)=180\left(1-e^{-0.2t}\right)$

Part c. Step 2. To determine.

We have to graph the velocity function.

Part c. Step 3. Calculation.

We will use Desmos graphing calculator to graph $v\left(t\right)$.

We will replace t by x in order to graph on Desmos.

So, the required graph is:

Part d. Step 1. Given.

Given that the downward velocity of the sky diver at time t is given by:

$v\left(t\right)=180\left(1-e^{-0.2t}\right)$

Part d. Step 2. To determine.

We have to find the terminal velocity.

Part d. Step 3. Calculation.

From part c, the horizontal asymptote is at $v\left(t\right)=180$. So as$t\to \infty$ then$v\left(t\right)$ approaches 180. 180 is also the maximum velocity.

It means the terminal velocity is 180 ft/s.