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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 velocity with respect to time.
The given equation is [tex]F(x)=k_1xk_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 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:
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!!
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_1xk_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*x115*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 xvector, 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.*xk2.*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.*xk2.*x.*exp(c.*x.^2))/m;
figure, plot(x,F,'g')
title('Force with respect to x')
I guess my problem can be stated another way:
How do I parametrize from x to t?
Thanks in advance!!
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