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Hello awesome physics people!

Someone asked me for help on their first year physics homework, and I couldn't really solve it. This kept bugging me, because I should know how this works by now :P

See attachment for the full problem statement. Basically, a bow is strung with an arrow, and what they want to know is, among other things, the graph for the

The given equation is [tex]F(x)=-k_1x-k_2xe^{cx^2}[/tex].

Other equations I've used:

[tex]E_{pot}=E_{kin}=1/2*mv^2[/tex]

[tex]v_{terminal}=sqrt{(2*E_{pot})/m_{arrow}}[/tex]

I've come as far as the calculation of the terminal velocity of the arrow, however, after that I need to come up with an equation of velocity with respect to time, whilst I have only the acceleration with respect to

I've implemented my partial solution in MATLAB, here is my code:

I've included the plot as another attachment.

I guess my problem can be stated another way:

Thanks in advance!!

Someone asked me for help on their first year physics homework, and I couldn't really solve it. This kept bugging me, because I should know how this works by now :P

## Homework Statement

See attachment for the full problem statement. Basically, a bow is strung with an arrow, and what they want to know is, among other things, the graph for the

**velocity with respect to time**.## Homework Equations

The given equation is [tex]F(x)=-k_1x-k_2xe^{cx^2}[/tex].

Other equations I've used:

[tex]E_{pot}=E_{kin}=1/2*mv^2[/tex]

[tex]v_{terminal}=sqrt{(2*E_{pot})/m_{arrow}}[/tex]

## The Attempt at a Solution

I've come as far as the calculation of the terminal velocity of the arrow, however, after that I need to come up with an equation of velocity with respect to time, whilst I have only the acceleration with respect to

**distance**from the equilibrium position of the bow. If it were linear I would be able to use a constant acceleration, but seeing as the acceleration is dependent on the distance from equilibrium, I don't really know what to do.I've implemented my partial solution in MATLAB, here is my code:

Code:

```
%% First I let MATLAB calculate the initial x:
solve('-390*x-115*x*exp(2.3*x^2)=350','x');
% The outcome of this: xin=-0.62035062944153550757039658657226, or -0,620m
%% Now for the numerical stuff
k1=390;
k2=115;
c=2.3;
Fmax=350;
x=-0.620:0.01:0;
% This is the x-vector, I initialized it from the initial x until x=0, when
% the arrow leaves the string.
dt=0.01; % This is the timestep
t=0:dt:5; % Here I construct a vector t from 0 to 5 with steps of dt
m=34*10^-3; % This is the mass of the arrow
% The following is the given formula:
F=-k1.*x-k2.*x.*exp(c.*x.^2);
% To get the energy stored up in the spring, we need only to integrate F
% over x:
Epot=trapz(F,x); % This gives Epot = 146.4474 J
% As all of the potential energy will be transferred into kinetic energy of
% the arrow, according to Epot = Ekin = 1/2*mv^2, we can calculate the
% terminal velocity of the arrow:
vterm=sqrt((2*Epot)/m); % This gives vterm = 92.8146 m/s
% Assuming all of the force gets transferred to the arrow, and because
% F=m*a, we can calculate the acceleration of the arrow as function of x
% simply by dividing by the mass of the arrow:
a=(-k1.*x-k2.*x.*exp(c.*x.^2))/m;
figure, plot(x,F,'g')
title('Force with respect to x')
```

I've included the plot as another attachment.

I guess my problem can be stated another way:

*How do I parametrize from x to t?*Thanks in advance!!

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