How can I integrate an equation of acceleration vs position to position vs time.

An uncontrolled automobile traveling at 45 mph strikes a highway crash barrier square on. After initially hitting the barrier, the automobile decelerates at a rate proportional to the distance x the automobile has moved into the barrier; specifically, a=(-60)*sqrt(x) where a is experess as ft/s^2 and x as ft. Develope the plots for position vs time, velocity vs time and acceleration vs time for a vehicle initallity traveling at 40 mph, 50 mph and 60 mph.

So far I set the (-60)*sqrt(x) to v*(dv/dx) then rearranged the equation to get vdv=(-60)*sqrt(x). Then I integrated the dv side from Vinitial to Vfinal and the dx from 0 to Xfinal to find velocity in terms of position. Vfinal=sqrt(-80x^(3/2)+(Vi^2)).

I then set Vfinal equal to dx/dt so that sqrt(-80*xfinal^(3/2)+(Vinital^2))=dx/dt --> dt=dx/sqrt(-80x^(3/2)+(Vi^2)). Then I need to integrate the dt side from 0 to Tfinal and dx from 0 to Xfinal but doing that in matlab is where I get stuck. Then I have no clue how I can use matlab to rearrange the equation so that its position in terms of time and not time in terms of position.

Answer 1

[Raghavendra Ragipani]

Hi,

You can simplify the equations as follows:

dv/dt = (-60)*sqrt(x)

dx/dt = v

with initial conditions t=0: x=0 and v=?

Integrate two dependent variables (x,v) using ode45.

You can take help from matlab help on ode45 for further instructions.

cheers,

raghav