Chapter 4, Problem 7SP

A skydiver has a weight of 850 N. Suppose that the air-resistive force acting on the diver increases in direct proportion to his velocity such that for every 10 m/s that the diver’s velocity increases, the force of air resistance increases by 100 N.

1. a. What is the net force acting on the skydiver when his velocity is 30 m/s?
2. b. What is the acceleration of the diver at this velocity?
3. c. What is the terminal velocity of the skydiver?
4. d. What would happen to the velocity of the skydiver if for some reason (perhaps a brief down draft) his velocity exceeded the terminal velocity? Explain.

To determine

The net force acting on the skydiver of weight $850\text{N}$ if the air resistive force will increase by $100\text{N}$ for every $10\text{m}/\text{s}$ increase in the velocity.

Answer to Problem 7SP

The net force acting on the skydiver of weight $850\text{N}$ if the air resistive force will increases by $100\text{N}$ for every $10\text{m}/\text{s}$ increase in the velocity is $550\text{N}$.

Explanation of Solution

Given info: The weight of the skydiver is $850\text{N}$ and increase in the air resistive force for every $10\text{m}/\text{s}$ increase in the velocity is $100\text{N}$.

If the air resistive force will increases by $100\text{N}$ for every $10\text{m}/\text{s}$ increase in the velocity,$300\text{N}$ will increase for a $30\text{m}/\text{s}$ increase in the velocity.

Write the net force acting on the skydiver.

$F_{\text{net}}=F_{\text{upward}}+F_{\text{downward}}$

Take $F_{\text{upward}}$ as negative since it is acting in the upward direction.

The only force acting downward is the gravitational force and force acting in the upward direction is the air resistive force.

Therefore substitute $850\text{N}$ for $F_{\text{downward}}$ and $-300\text{N}$ for $F_{\text{upward}}$ in the above equation to gjet $F_{\text{net}}$.

$\begin{array}{c}{F}_{\text{net}}=850\text{N}-300\text{N}\\ \text{=550}\text{N}\end{array}$

Thus the net force is in the downward direction.

Conclusion:

Thus, the net force acting on the skydiver of weight $850\text{N}$ if the air resistive force will increases by $100\text{N}$ for every $10\text{m}/\text{s}$ increase in the velocity is $550\text{N}$.

To determine

The acceleration of the diver at the velocity.

Answer to Problem 7SP

The acceleration of the diver at the velocity is $6.34\text{m}/{\text{s}}^{\text{2}}$.

Explanation of Solution

Given info: The mass of the man is $85\text{kg}$ and acceleration of the man is $1.3\text{m}/{\text{s}}^{\text{2}}$.

Write the expression for the mass of the diver.

$m=\frac{W}{g}$

Here,

$m$ is the mass of the diver

$g$ is the gravitational acceleration

Substitute $850\text{N}$ for $F_{\text{net }}$ and $9.8\text{m}/{\text{s}}^{\text{2}}$ for $g$ in the above equation to get $m$.

$\begin{array}{c}m=\frac{850\text{N}}{9.8\text{m}/{\text{s}}^{\text{2}}}\\ =86.73\text{kg}\end{array}$

Write the expression for the net force on the diver.

$F_{\text{net}}=ma$

Here,

$F_{\text{net}}$ is the net force acting on the diver

$a$ is the acceleration of the diver

Rearrange the above equation to get $a$.

$a=\frac{F_{\text{net}}}{m}$

Substitute $86.73\text{kg}$ for $m$ and $550\text{N}$ for $F_{\text{net}}$ in the above equation to get $F_{\text{net}}$.

$\begin{array}{c}a=\frac{550\text{N}}{86.73\text{kg}}\\ =6.34\text{m}/{\text{s}}^{\text{2}}\end{array}$

Conclusion:

Thus, the acceleration of the diver at the velocity is $6.34\text{m}/{\text{s}}^{\text{2}}$.

To determine

The terminal velocity of the skydiver.

Answer to Problem 7SP

The terminal velocity of the skydiver is $85\text{m}/\text{s}$.

Explanation of Solution

Given info: increase in the air resistive force for every $10\text{m}/\text{s}$ increase in the velocity is $100\text{N}$.

The diver will attain terminal velocity if the upward force is equal to the weight of the diver.

The weight of the diver is $850\text{N}$. This force is acting downward. This force can be balanced by the upward force of magnitude equal to $850\text{N}$.

Since increase in the air resistive force for every $10\text{m}/\text{s}$ increase in the velocity is $100\text{N}$. Therefore he will attain an air resistive force of 850 N only if velocity of the diver reaches $85\text{m}/\text{s}$.

Therefore at this velocity upward force becomes $8.5$ times the force for initial $10\text{m}/\text{s}$. Therefore terminal velocity of the skydiver is equal to $85\text{m}/\text{s}$.

Conclusion:

Thus, the terminal velocity of the skydiver is $85\text{m}/\text{s}$.

To determine

What would happen to the velocity of the skydiver if for some reason his velocity exceeded the terminal velocity?

Answer to Problem 7SP

This will cause windburn due to increase in the frictional force and sudden change in velocity to the terminal velocity.

Explanation of Solution

Terminal velocity is the uniform velocity of falling body in a viscous medium if the upward force is equal to downward velocity.

If the velocity is exceeded the terminal velocity, it will cause increase in the resistive force and dissipation of energy in the form of heat occurs. This may create wind burn. This resistive for reduce the velocity to terminal velocity.

Conclusion:

Thus, exceeding velocity more than terminal velocity will cause windburn due to increase in the frictional force and sudden change in velocity to the terminal velocity.