The parabola $\mathcal P$ given by equation $y=x^2$ is rotated some acute angle $\theta$ clockwise about the origin such that it hits both the $x$ and $y$ axes at two distinct points.  Suppose the length of the segment $\mathcal P$ cuts the $x$-axis is $1$.  What is the length of the segment $\mathcal P$ cuts the $y$-axis?
1. **Rotation and Intersection Points:**
   - The parabola \( y = x^2 \) is rotated by an acute angle \(\theta\) about the origin.
   - After rotation, the parabola intersects the \(x\)-axis at two distinct points. Let these points be \((a, 0)\) and \((b, 0)\) with \(a < b\).
   - Given that the length of the segment on the \(x\)-axis is 1, we have \(b - a = 1\).

2. **Coordinate Transformation:**
   - Let the new coordinates after rotation be \((u, v)\).
   - The transformation equations for rotation by angle \(\theta\) are:
     \[
     u = x \cos \theta + y \sin \theta
     \]
     \[
     v = -x \sin \theta + y \cos \theta
     \]
   - Substituting \(y = x^2\) into these equations, we get:
     \[
     u = x \cos \theta + x^2 \sin \theta
     \]
     \[
     v = -x \sin \theta + x^2 \cos \theta
     \]

3. **Intersection with the \(x\)-axis:**
   - For the parabola to intersect the \(x\)-axis, \(v = 0\):
     \[
     -x \sin \theta + x^2 \cos \theta = 0
     \]
     \[
     x(-\sin \theta + x \cos \theta) = 0
     \]
     \[
     x = 0 \quad \text{or} \quad x = \frac{\sin \theta}{\cos \theta} = \tan \theta
     \]
   - The non-zero intersection point is \(x = \tan \theta\).

4. **Length of Segment on \(x\)-axis:**
   - The length of the segment on the \(x\)-axis is given as 1:
     \[
     \tan \theta - 0 = 1 \implies \tan \theta = 1 \implies \theta = \frac{\pi}{4}
     \]

5. **Intersection with the \(y\)-axis:**
   - For the parabola to intersect the \(y\)-axis, \(u = 0\):
     \[
     x \cos \theta + x^2 \sin \theta = 0
     \]
     \[
     x(\cos \theta + x \sin \theta) = 0
     \]
     \[
     x = 0 \quad \text{or} \quad x = -\frac{\cos \theta}{\sin \theta} = -\cot \theta
     \]
   - The non-zero intersection point is \(x = -\cot \theta\).

6. **Length of Segment on \(y\)-axis:**
   - Substituting \(\theta = \frac{\pi}{4}\):
     \[
     \cot \theta = \cot \frac{\pi}{4} = 1
     \]
   - The intersection point on the \(y\)-axis is \((0, -1)\).
   - The length of the segment on the \(y\)-axis is:
     \[
     \sqrt{(0 - 0)^2 + (1 - (-1))^2} = \sqrt{0 + 4} = 2
     \]

The final answer is \(\boxed{2}\).