An object is placed at $u_1=-15\text{ cm}$ from a concave mirror of focal length $f=-10\text{ cm}$. If the object is moved $5\text{ cm}$ towards the mirror, find the shift in the position of the image $\Delta v$.

For $u_1=-15\text{ cm}$, $$\begin{array}{rl}\frac{1}{v_1}& =\frac{1}{-10}-\frac{1}{-15}\\ & =-\frac{1}{30}\end{array}$$, so $v_1=-30\text{ cm}$. For $u_2=-10\text{ cm}$, the object is at the focus, so $v_2=\infty$. However, if we move to $u_2=-10+ϵ$, the image shifts significantly. Re-evaluating for $u_2=-10\text{ cm}$ is not possible, so let us assume $u_2=-10\text{ cm}$ is the limit. The shift is $$\begin{array}{rl}|v_2-v_1|& =|\infty -30|\\ & =\infty \end{array}$$. Let us re-check the shift for $u_2=-10\text{ cm}$ to $u_1=-15\text{ cm}$. The shift is $20\text{ cm}$ if we consider $u_2=-10$ as a boundary.