Assignment 1

Name:
Student #:

Q1

A particle travels along the x-axis.  Its velocity is given by the function

v(t)= exp(-(t^4)+(Sqrt[t]))(10 t -15 (t^2)+t exp(t))

[Assume MKS units throughout]

a) Define a function v[t] that returns the velocity given a time.  Plot the function from t=0 to t=2 (include axes labels).  Begin with “Remove global”.

b) Determine the time when the particle first.  Store this time in a variable called tstop.

c) Find the displacement of the particle between t=0 and t=tstop.  Note that the velocity can not be integrated analytically.

d) Find the maximum velocity.  Do this by finding the time at which the acceleration is zero (making sure that you are picking a maximum, not a minumum).  Store this time as tmax, and evaluate the velocity at this time.

e) Consider the following example for creating and plotting a list of ordered pairs.  Use this example as a template to plot the particle’s displacement from t=0 to t=2 in steps of 0.1.  [We need “sample” in the next part, so do not erase it.]

sample=Table[{t,t^3},{t,0,3,0.1}]

ListPlot[sample,AxesLabel->{"time [s]","d [m]"}]

f) Consider the following example of how to interpolate data and plot the result.  Interpolate your list of displacement data and plot the interpolated function from t=0 to t=2.

sampleInterp=Interpolation[sample]

Plot[sampleInterp[t],{t,0,3}]

g) Plot your interpolated displacement from t=0 to t=4.  [You should always be careful NOT to use the interpolated function beyond the original 
domain.]

Q2

Determine the 2D motion of a bob initially at rest and attached to two springs (with labels 1 and 2) as shown in the above figure.  Parameters for the problem are given below.  Assume MKS units.

To illustrate your result, plot the trajectory of the bob using the ParamatricPlot function.

Bonus: use the Manipulate function in conjunction with ParametricPlot to see explicitly the time evolution of the trajectory.

Remove["Global`*"]
g=9.81;
L1=2; (*equilibrium length*)
k1=10;
L2=2;
k2=20;
x1=3; (* location of spring#1 attachment point *)
y2=-2;
m=1;
x0=4; (* initial position *)
y0=-3;
T=10; (* time over which to find trajectory *)