Section 13.4 (6th edition)

load(vect);

load(draw);

Example 7: (a) Find the velocity, acceleration and speed of the position
[t^2,t^2,t^4]. (b) Find the tangential and normal components of acceleration.

rr: [t^2,t^2,t^3];

vv: diff(rr,t);

aa: diff(vv,t);

t0: 1;

draw3d(xlabel="X", ylabel="Y", line_width=2, head_length=0.02,
    xrange=[0,2], yrange=[0,2], zrange=[-2,2],
    color=1, parametric(rr[1],rr[2],rr[3],t,-2,0),
    color=2, parametric(rr[1],rr[2],rr[3],t,0,2),
    color=3, vector(at(rr,t=t0),at(vv,t=t0)),
    color=4, vector(at(rr,t=t0),at(aa,t=t0)));

cc: express(vv ~ aa);

at: (vv.aa)/sqrt(vv.vv);

an: sqrt(cc.cc)/sqrt(vv.vv);

*Exercise 10: Find the velocity, acceleration and speed of the position
[2*cos(t),3*t,2*sin(t)]

rr: [2*cos(t),3*t,2*sin(t)];

vv: diff(rr,t);

aa: diff(vv,t);

trigsimp(sqrt(vv . vv));

*Exercise 16: Find the velocity and position given the acceleration [2,6*t,12*t^2]
and given the initial velocity [1,0,0] and the initial position [0,1,-1]

aa: [2,6*t,12*t^2];

vv: [1,0,0]+integrate(aa,t,0,t);

rr: [0,1,-1]+integrate(vv,t,0,t);

*Exercise 18:  Find the velocity and position given the acceleration [t,exp(t),exp(-t)]
and given the initial velocity [0,0,1] and the initial position [0,1,1].
Then draw the path of positions.

aa: [t,exp(t),exp(-t)];

vv: [0,0,1]+integrate(aa,t,0,t);

rr: [0,1,1]+integrate(vv,t,0,t);

wxdraw3d(color=red, parametric(rr[1],rr[2],rr[3],t,0,6));

*Exercise 34: Find the tangential and normal components of acceleration for
the position [1+t,t^2-2*t,0]. And draw the path of positions.

rr: [1+t,t^2-2*t,0];

drr: diff(rr,t);

d2rr: diff(rr,t,2);

a_T: drr.d2rr/sqrt(drr.drr);

cross: express(drr~d2rr);

a_N = sqrt(cross.cross)/sqrt(drr.drr);

wxdraw3d(color=red, parametric(rr[1],rr[2],rr[3],t,0,6));