= Solution
The <arc length> of this <parametric curve> is the integral of its speed. <Differentiation> gives $x'(t)=at\cos t$ and $y'(t)=at\sin t$, so
$$
\sqrt{x'(t)^2+y'(t)^2}=|a|t\qquad(0\leq t\leq1),\qquad \boxed{\ell=\int_0^1|a|t\,dt=\frac{|a|}{2}.}
$$
For a positive scale parameter this is $a/2$. The absolute value is necessary when $a$ is allowed to be any real constant; $a=0$ gives a degenerate curve of zero length. The nondegenerate curve is an <involute> of a circle.
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