Question #43132

This question tests the synthesis of orbital mechanics (areal velocity) and optics (variable magnification).

The primary trap lies in the time-varying nature of the magnification $m_T$.

Students often attempt to find the image velocity components directly, which requires the unknown radial velocity $\dot{x}_o$ to compute $\dot{m}_T$.

The physical insight is recognizing that the image position vector scales exactly as $\vec{r}_i = m_T \vec{r}_o$.

When computing the areal velocity via the cross product $\vec{r}_i \times \vec{v}_i$, the term involving $\dot{m}_T \vec{r}_o$ is parallel to $\vec{r}_o$ and vanishes, leaving $\frac{dA_i}{dt} = m_T^2 \frac{dA_o}{dt}$.

Furthermore, students must correctly apply Newton's lens equation to find $m_T = -f/x$ and remember the $1/2$ factor in Kepler's areal velocity formula $\frac{dA}{dt} = \frac{L}{2m}$.

Common mistakes include assuming areal velocity is conserved for the image (Option B), forgetting to square the magnification (Option C), or forgetting the $1/2$ factor in the areal velocity formula (Option D).