How to parameterise a parabola with a particle running on it at a given speed, say 10 m/s.

Question

If I use t=x, it will dictate a uniform velocity in the x-direction, and eventually violate the constraint. Could use some hints. [edit: ... at a given speed, not velocity, my apologies].

Answer 1

Hint : velocity of the particle = $$\sqrt{x'^2(t)+y'^2(t)}$$ and you should know the expression of $y$ because $y(t)=f(x(t))$ with $f$ your parabola equation

Answer 2

The answer depends on what you consider its velocity. As you give a scalar quantity and no direction, I would assume the constraint is about the length of the velocity vector.

$$ 10 = \lVert v \rVert = \sqrt{ v\cdot v } $$

A parametrization of the parabola is $$ u = (f(t), f(t)^2)^T $$ it leads to $$ v = \dot{u} = (\dot{f}(t), 2\,f(t)\, \dot{f}(t))^T $$ and \begin{align} 10 &= \sqrt{(\dot{f}(t))^2+ 4 \,f(t)^2 \, (\dot{f}(t))^2} \\ &= \left\lvert \dot{f}(t) \right\rvert \sqrt{1 + 4 \, f(t)^2} \end{align} For positive $\dot{f}$ this means $$ 10 \, dt = \sqrt{1 + 4 \, f(t)^2} \, df $$ which according to WolframAlpha leads to $$ t = \frac{1}{20} f(t) \sqrt{4 f(t)^2+1}+\frac{1}{40} \sinh^{-1}(2 f(t)) + C $$ And I fear this leaves only a numerical calculation of $f = t^{-1}$, like in this WA result.