Let's say I have the parametric function$$f(t)=5\text i+\cos t\text j+(1+\sin t)\text k$$Where $t$ ranges from $-\frac \pi2$ to some number, say $a$. I want to be able to connect $f(t)$ with another function $g(t)$ that takes the form of a parabola $$g(t)=5\text i+t\text j+\frac 4{25}t(50-t)\text k$$ such that the piecewise function$$F(t)=\begin{cases}f(t)\qquad-\frac \pi2\leq t\leq a\\\\g(t)\qquad a\leq t\leq\infty\end{cases}$$ is both continuous and differentiable at $t=a$. How do I go about finding the $a$ value?
I tried finding the curvature of both equations to set them equal to each other. The curvature of $f(t)$ is simply one while the curvature of $g(t)$ is$$K_{g(t)}=\frac {5000}{\left[64(t-25)^2+625\right]^{3/2}}$$However, there is no value of $t$ according to Wolfram Alpha that has the curvature of $g(t)$ equal to one. Is there any way for me to actually solve for $a$ and $g(t)$?