# Shooting Arrows - Nonlinear Spring Problem

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

## 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 $$F(x)=-k_1x-k_2xe^{cx^2}$$.

Other equations I've used:

$$E_{pot}=E_{kin}=1/2*mv^2$$
$$v_{terminal}=sqrt{(2*E_{pot})/m_{arrow}}$$

## 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?

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gneill
Mentor
Your value shown for xin looks a bit suspicious. Verify your solution.

For the V vs T plot you're going to have to do a numerical integration of the trajectory. Use whatever algorithm you think appropriate, such as the Leapfrog method, or Euler's method, or Verlet, or even Runge-Kutta (probably overkill). Some web time will fill you in on those.

They all boil down to using the force at a given position to determine an acceleration to apply over the next small time interval. Use it to update the velocity and position, rinse, repeat.

Thanks for your reply gneill! I see, so there is no analytical method of converting distance into time when working with a nonlinear system?

My x_initial doesn't seem suspicious to me, loading a bow by pulling back 62 cm is plausible, right? The minus sign is explicable by the definition of the x-axis in the problem statement.

I'll try to implement this Frogleap integration method, thanks again!

gneill
Mentor
Thanks for your reply gneill! I see, so there is no analytical method of converting distance into time when working with a nonlinear system?
It depends upon the functions involved. If you run into a transcendental function you're basically hosed In this case the x2 in the exponent of the exponential is a major stumbling block for a solution involving the normal basis set of functions that we're used to.

My x_initial doesn't seem suspicious to me, loading a bow by pulling back 62 cm is plausible, right? The minus sign is explicable by the definition of the x-axis in the problem statement.
I get a different value when I find the root of the function. In particular, I find x = -0.559 m.

I'll try to implement this Frogleap integration method, thanks again!
Sounds like a plan!

Yay, it worked!

I implemented the Leapfrog algorithm, and it worked! At first I got a sine wave (which isn't strange seeing as I basically modelled a spring), so I made the calculation stop as soon as x became positive. A bit of a hack solution, also because the rest of the vectors are zeros after that (that explains the sudden straight line in the graphs), but I didn't feel like polishing the code. I also used the erroneous value of -0.62 for x_initial, as it didn't really matter for this implementation (but thanks for checking anyways!)

Here's my code:

Code:
%% Now attempting a Leapfrog integration
dt=0.001;
t=0:dt:0.013;
x_in=-0.620;
x_t=zeros(1,length(t));
v_t=zeros(1,length(t));
a_t=zeros(1,length(t));
x_t(1)=x_in;
v_t(1)=0;
a_t(1)=(-k1.*x_in-k2.*x_in.*exp(c.*x_in.^2))/m;

i=2;
while x_t(i-1)<0

x_t(i)=x_t(i-1)+v_t(i-1)*dt+0.5*a_t(i-1)*dt^2;
a_t(i)=(-k1*x_t(i)-k2*x_t(i).*exp(c*x_t(i).^2))/m;
v_t(i)=v_t(i-1)+0.5*(a_t(i-1)+a_t(i)).*dt;
i=i+1;
end

hold all
subplot(2,2,1), plot(t,v_t,'b'), title('Velocity with respect to time')
subplot(2,2,2), plot(t,x_t,'r'), title('Distance with respect to time')
subplot(2,2,3), plot(t,a_t,'g'), title('Acceleration with respect to time')
And my graphs are attached. Cheers gneill :)

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