8. Motion in One Dimension with Air Drag

The purpose of this lab is to analyze how changing force affects motion in one dimension.

Introduction: If a changing force (which causes a changing acceleration) is applied to an object then the object Fnet will be different at every time.

Ø  An object under the influence of Drag is one example. Two basic relationships can be used to analyze the object.              Vnew = Vbold + aavg

1.       By unit analysis we can show that the equation above is valid.

2.       We use aavg  in equation 1 because a is changing therefore taking the average is gives a more accurate value.

3.       We come up with an analogous equation relating Ynew to Yold.

                                       Vnew=VoldΔt+ aavg Δt

                                                                               2

4.       The benefit of choosing a small Δt is that the average of acceleration is closer to the actual acceleration problem.

 

Ø  Since the acceleration depedends on the forces acting on the oblject, we must specify precisely what these forces are. I our problem the drag force will be assumed to depened on the first power of the velocity, and can be written as  F_D = - Kv

Where K is proportionally constant.

 

1.       Motion diagram of the object falling down.

 

a= F / m = D - G = (-kv-mg) / m = - (g + kv / m )


b. Give the condition for the object at terminal velocity (a=0). Using this condition solve for k.

a= 0, So net force is 0, mg = kv_t , k= mg / v_t

c. Substitute k into your expression for the acceleration from part a.

a = - (g + kv / m )
k = mg / v_t So, a = -g (1+ v / v_t )


2. Open the spreadsheet in the Physics Apps file called air drag. Describe what the spreadsheet is calculating.

a. What are the assumption? (i.e. initial values? )

t_o = 0s , g = -9.8m/s^2 , initial velocity = 20m/s , terminal velocity = -40m/s , △t = 0.1s

b. What is v-halfstep?

V-halfstep is the average velocity in 0.1s.

c. What is the a-halfstep?

A-halfstep is the average acceleration in 0.1s.


3. Using the graph paper provided, draw scales graphs of position vs. time, velocity vs. time and acceleration vs. time for the object no air drag.

position vs. time graph when there is no drag

p= -4.9t^2 + 20t +1000

velocity vs. time graph when there is no drag

v= -9.8t + 20

Make predictions on another sheet of graph paper about how position and velocity graphs would change if you include air drag (D= -kv).

position vs. time graph when drag is equal to -kv

velocity vs. time graph when drag is equal to -kv

Now look at a drag force that is dependent on the square of the velocity. Assuming a drag force,
F_D= | kv^2 |, find the new formula for the acceleration.

a= F / m = D - G = (kv^2 - mg) / m
mg = k v_t^2 , k = mg / v_t ^2
So, a= [(v^2/ v_t^2) -1] g

position vs. time graph when drag is equal to |kv^2|

velocity vs. time graph when drag is equal to |kv^2|

Conclusion:

In this lab, we analyzed how changing force affects motion in one dimension. We plotted v-t and p-t graphs in three cases: with no drag; drag is equal to -kv and drag is equal to | kv^2 |.