Problem with accelerating objects with drag

[bassbrotha]

hey guys, I am new here and this is my first post, but i need some help.

the concept i am trying to explain mathematically is the acceleration a skateboarder (150kg) will have going down a hill (5 degrees bellow the horizontal) completely straight (2 dimensional), including air drag. to make matters more simple, let's assume the rolling friction is non existent.

i understand to find the acceleration, i have to get the sum of all forces. so therefore

Fnet = Fgx - Fd (fgx being the gravity component and the fd being drag)

i understand that Fgx you can find simply by plugging numbers into Fgsin(pheta)= Fgx

the thing that is stumping me is the acceleration the drag applies to the object. i understand the formula F= 0.5 Cd p v^2 A (i have my given Cd, p and A values as 0.7, 1.1897 and 0.8). but the problem that i am having is that this formula only shows the instantaneous velocity, therefore i do not know how to obtain a net acceleration. So my question is, how do you find the net force (in order to find the net acceleration)

i am a grade 11 student taking 12 physics right now, so i do not understand calculus, so if any of it is used, if you can be clear it would be much appreciated

and lastly, i was thinking that graphing it would help possibly to understand the relationship (that is since the forces are changing).

thanks for your help guys in advance :)

 

[AlephZero]

i understand to find the acceleration, i have to get the sum of all forces. so therefore

Fnet = Fgx - Fd (fgx being the gravity component and the fd being drag)

i understand that Fgx you can find simply by plugging numbers into Fgsin(pheta)= Fgx

the thing that is stumping me is the acceleration the drag applies to the object. i understand the formula F= 0.5 Cd p v^2 A (i have my given Cd, p and A values as 0.7, 1.1897 and 0.8). but the problem that i am having is that this formula only shows the instantaneous velocity, therefore i do not know how to obtain a net acceleration. So my question is, how do you find the net force (in order to find the net acceleration)

You have found the net (resultant) force, so you find the acceleration be dividing the force by the mass.

If you don't know calculus yet, you can only find the instanteous force and acceleration for a particular velocity. You can't find the average acceleration down the hill, which might be more interesting.

You can find the maximum velocity (or terminal velocity). If the velocity is constant, then the acceleration is 0, so Fnet = 0 or Fge = Fd. You can solve that equation to find v.

You can get an approximate numerical answer for the speed down the slope. Start with x = 0 and v = 0 and find the intial acceleration. (That will be Fge/m). Then, assume the acceleration is constant for a short time, say 1 second. You can use the formulas for constant acceleration to find the distance and time after 1 second.

v = v0 + at
x = x0 + v0 t + (1/2)at²

Then calculate the new acceleration at 1 second (which will be a bit smaller than the acceleration at the start) and find v and x after 2 seconds. etc.

You will need a computer (either write your own computer program, or use a spreadsheet), or at least a programmable calculator, to do this.

 

[bassbrotha]

thanks for your help! I had a feeling that this would result in having to crunch the numbers with individual data peices and graph them to show the relationship.

even though its not the outcome i desired, i thank you for your help of clarification.

 

[AlephZero]

Well, you now have a motivation for wantiing to learn calculus

(But it's only fair to warn you this isn't an "easy" calculus problem, so don't expect to be able to solve it as soon as you start your first calculus course.)

 

[rcgldr]

I had a feeling that this would result in having to crunch the numbers with individual data peices and graph them to show the relationship.

That is the normal process in many cases, since a direct solution (equation) often can't be determined (situation too complex to use direct integration (calculus)).

Numerical integration involves a lot of steps, but the basic algorithm is not difficult to understand. One way to estimate velocity is to use "Euler" approximation with a small Δt (time step):

new_velocity = old_velocity + old_acceleration x Δt

This can be improved by using average acceleration

average_acceleration = 1/2 (old_acceleration + new_acceleration)

but you need to use an iterative method that will allow new_acceleration to be estimated accurately.

There is an example of this in post 2 of this thread with a minor correction in post 4:

thread_469897_post 2