The following procedure describes the path of a projectile fired from the origin with a launch angle of alpha and a launch velocity of u.

z := proc (x) options operator, arrow; tan(alpha)*x-(1/2)*g*x^2/(u^2*cos(alpha)^2) end proc

The projectile's path must pass through the point [a, h], both values greater than zero.

Can an expression be found for the minimum value of u and its related value for alpha?

The object in the above display consists of a thin red disk and attached thin blue disk. They have radii 1.0 and 0.2 and masses 1.0 and 0.2 respectively.

Starting from rest at the above position they are given an initial angular velocity of Omega radians/second which starts the object rolling along the positive x axis.

If Omega is great enough that, at some time during the roll, the vertical component of the centrifugal force on the blue disk exceeds the downward force of gravity on the object, will the object begin a spinning projectile motion above the x axis?

Assuming both disks have negligible thickness, what Physics laws will govern the rolling and (possible) spinning projectile motions and what mathematics describes these motions? 

 ODE:= -2 sin(1/2 theta(t)) cos(1/2 theta(t)) (diff(theta(t),t)^2-9.8000 sin(theta(t))-(150+4 sin(1/2 theta(t))^2) (diff(theta(t),t,t)=0:

 ICS:=  theta(0) = Pi/6, D(theta)(0) = DthetaZero:

The worksheet below shows a river whose velocity is a function of the distance from its mid point to either bank.

In this worksheet, as an example, a boat of fixed speed crosses the river heading at all times towards its destination which is directly across the river from its starting point. The boat's path is animated.

I presume an experienced boat captain could constantly vary his heading so as to cross in a minimum time.

Can a function for this minimum crossing time heading be found and the worksheet altered to show the minimum time path?

  Rivercrossing.mw

plot3d of procedure Sievert correctly displays the constant curvature Sievert surface, but the procedure uses the deprecated command evalm.

What Maple 2016 statement(s) would create the same value of X in Sievert?

Sievert := proc (B)

local a, b, denom, m, X;

a := sinh(B)*u; b := cosh(B)*v;

denom := sinh(B)*((cosh(2*a)-cos(2*b))*cosh(2*B)+2+cosh(2*a)+cos(2*b));

m := cosh(B)*[sinh(a), sin(b)*cos(v), sin(b)*sin(v)]+[0, -cos(b)*sin(v), cos(b)*cos(v)];

X := evalm([u, 0, 0]-8*cosh(B)*cosh(a)*m/denom);

end proc:

plot3d(Sievert(.75), u = -2.5 .. 2.5, v = -10.5 .. 10.5, scaling = constrained, grid = [30, 100], style = patch, shading = xy, lightmodel = light3, orientation = [-3, 140], title = "Sievert's surface", titlefont = [Courier, bold, 14]);