Assignment 3

Name:
Student #:

1.

The potential energy of a particle is described by the one-dimensional function given below.  Assume MKS units throughout.

Remove["Global`*"]

u[x_]:=20Exp[((-(((x-3))^2))/2)]+15 Exp[((-(((x+4))^2))/2)]+10 Exp[((-(((x+0.1))^2))/2)]+(10^(-4)) (x^6)
pot=Plot[u[x],{x,-8.1,8.1},PlotStyle->Red]

a)

If the particle starts from rest at x=-8.0, at what positive value of x should the particle start to reverse direction according to conservation of energy?

b)

Plot the force on the particle as a function of x.

c)

If a particle of mass 2.3 kg starts at rest at position x=-8.0, determine and plot the position and velocity of the particle for the first 30s.

d)

Plot the potential, kinetic and total energy of the particle as functions of time for the first 10s on one plot.  The total energy should be constant.

e)

Using the Manipulate command, produce an animation showing the particle’s position and energy time evolution, i.e., show the particle as a small disc travelling over the energy landscape.

Manipulate[Show[Plot[BLANK, {x, -8.1, 8.1}], ListPlot[{{X[t], BLANK}}, PlotStyle -> {Red, PointSize[0.02]}]], {BLANK, BLANK, BLANK}]

f)

Add a drag force of -0.2v and redo the calculation of the trajectory and the animation.  Also plot the potential, kinetic and total energy as functions of time for the first 30 s.

2.

Consider particles interacting with the pair potential 

V[r_]:= 4ϵ (((((σ/r)))^12)-((((σ/r)))^4)),

where r is the distance between the particles.  Take ϵ and σ as unity. At time t=0, there are three masses (Subscript[m, 1])=1, (Subscript[m, 2])=20 and (Subscript[m, 3])=2 with positions (Subscript[r, 1])={-3,1}, (Subscript[r, 2])={0,0} and (Subscript[r, 3])={3,-2}, and velocities (Subscript[v, 1])={2,0}, (Subscript[v, 2])={0,0} and (Subscript[v, 3])={-2, 2}.  Plot the trajectory of the particles on one graph up to (Subscript[t, f])=10.  Note that in this problem, all motion is confined to the x-y plane.

Remove["Global`*"]

sol = NDSolve[{eq1x, eq1y, in1x, in1y, in1vx, in1vy,eq2x, eq2y, in2x, in2y, in2vx, in2vy,eq3x, eq3y, in3x, in3y, in3vx, in3vy}, {x1, y1, x2, y2,x3,y3}, {t, 0, tf}][[1]];
X1[t_] = x1[t] /. sol;
Y1[t_] = y1[t] /. sol;
X2[t_] = x2[t] /. sol;
Y2[t_] = y2[t] /. sol;
X3[t_] = x3[t] /. sol;
Y3[t_] = y3[t] /. sol;

Also show a movie of the motion of the three particles.   Use the following line of code as a guide.

Manipulate[ListPlot[{{{Sin[t], Cos[2 t]}}, {{Sin[t/2] + 1, Cos[t] + 1}}}, PlotStyle -> {{Black, PointSize[0.03]}, Red}, PlotRange -> {{-4, 12}, {-3, 21}}], {t, 0, 10}]