What Is the Terminal Velocity of a 70 kg Sky Diver?

[link1388]

Homework Statement

A sky diver with a mass of 70 kg jumps from an aircraft. The aerodynamic drag force acting on
the sky diver is known to be FD =kV^2

, where k=.25 N.s^2/m^2

Determine the maximum speed of free fall for the sky diver and the speed reached after 100 m of fall. Plot the speed of the sky
diver as a function of time and as a function of distance fallen

Homework Equations

F=ma

Fg - kV^2 = ma

The Attempt at a Solution

Well I did find Velocity Max to be

mg -.25V^2 = ma

mg - .25V^2 = 0 (a=0 because of terminal velocity)

Vmax = 52.4 m/s

But I am really confused on finding V(x) so I can find the speed at 100m? : /

 

[edgepflow]

First find V(t). Apply Newton's 2nd law again:

Fnet = ma = m dv/dt

The forces acting on the sky diver, as you noticed are:

Fnet = mg - k v^2

Now you can solve this differential equation for V(t).

The position function, s(t), can then be found from: V(t) = ds(t) / dt.

 

[RTW69]

Great problem. The first integration is pretty straight forward but the second integration to find dx/dt=v is ugly as is the simplifications. I would suggest going to the Wolframalpha.com site for the math.

 

[link1388]

Ok I think I got the first integration but the second ones definitely going to be tricky lol Thanks.

 

[DeeNos]

Great job calculating the maximum velocity of the sky diver! To find the speed at 100m, we can use the equations of motion:

Vf^2 = Vi^2 + 2ad

where Vf is the final velocity, Vi is the initial velocity, a is acceleration, and d is distance.

In this case, we know the initial velocity (zero), the acceleration (due to gravity, which is constant at 9.8 m/s^2), and the distance (100m). So we can rearrange the equation to solve for Vf:

Vf = √(2ad)

Vf = √(2(9.8)(100))

Vf = √1960

Vf = 44.2 m/s

So the speed of the sky diver after falling 100m is 44.2 m/s.

To plot the speed as a function of time, we can use the equation:

V = gt

where g is the acceleration due to gravity and t is time. Since we know the acceleration and can assume the starting velocity is zero, we can plot the speed as a function of time using this equation.

To plot the speed as a function of distance fallen, we can use the equations of motion again, but this time solving for distance:

d = Vi*t + 1/2*a*t^2

Since we know the initial velocity and acceleration, we can plot the speed as a function of distance using this equation.

I hope this helps! Remember to always check your units and use the correct equations for the given situation. Good luck!