Why does solve not find the value of xp = sqrt(1-v^2) in this worksheet?

SpecialRelativity.mw

Reference: Special Relativity and Classical Field Theory. Authors: Leonard Susskind and Art Friedman

This solve command produces both unconditional and conditional solutions.

Is there a way to have it only output unconditional solutions, or, failing that, is there a way to select the unconditional solutions from its output?

solve({2*x+y+z = 3, x^2+y^2+z^2 = 3}, [x, y, z], real, parametric)

The linked worksheet displays the geodesic between two points on a surface z(x,y).

Surfacepath.mw

Gradient descent will find a stepped approximate path of fastest descent between the points, but is there a way to find a function defining an exact i.e. smooth path of fastest descent? If so, how can a precise time of fastest descent be determined?

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?