# Relation between Object and Image Velocity

## Relation between Object and Image Velocity

An image is the point of convergence or apparent point of divergence of rays after they interact with a given optical element. An object provides rays that will be incident on an optical element. The optical element reflects or refracts the incident light rays which then meet at a point to form an image. As in the case of objects, images too can be real or virtual.

Case (1): Differentiate equation $$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$$ with respect to time.

$$-\frac{1}{{{v}^{2}}}\frac{dv}{dt}-\frac{1}{{{u}^{2}}}\frac{du}{dt}=0$$.

$$-\frac{1}{{{v}^{2}}}{{V}_{im}}-\frac{1}{{{u}^{2}}}{{V}_{OM}}=0$$.

$$\frac{dv}{dt}={{V}_{im}}$$ = Velocity of image with respect to mirror.

$${{V}_{im}}=-\frac{{{v}^{2}}}{{{u}^{2}}}{{V}_{OM}}$$.

$$\frac{du}{dt}={{V}_{OM}}$$ = Velocity of object with respect to mirror.

Vim = – m²VOM.

The negative sign shows that if u is decreasing, v will increase, i.e. if real object approaches the mirror, its real image will recede from the mirror.

In this case |m| = 1, hence $$\left| \frac{dv}{dt} \right|\,\,<\,\,\left| -\frac{du}{dt} \right|$$.

When the object is at centre of curvature, $$\left| \frac{dv}{dt} \right|\,\,=\,\,\left| -\frac{du}{dt} \right|$$.

Case (2): Object moves between centre of curvature and focus.

In this case |m| = 1, hence $$\left| \frac{dv}{dt} \right|\,\,>\,\,\left| -\frac{du}{dt} \right|$$.

Case (3): Object moves between focus and pole of the mirror.

In this case image is virtual, hence $$\frac{1}{(+v)}+\frac{1}{(-u)}=\frac{1}{(-f)}$$.

$$\frac{dv}{dt}=\frac{{{v}^{2}}}{{{u}^{2}}}\frac{du}{dt}$$.

If u is decreasing, v will also decrease, i.e. if real object approaches mirror, image will also do so. As |m| > 1, speed of image will be greater than speed of object. Size of image for a small size object placed along principal axis.

Differentiating $$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$$ with respect to t, we get:

$$\frac{dv}{dt}=-{{m}^{2}}\frac{du}{dt}$$.

dv = -m²du.

$$\frac{du}{dt}$$ and $$\frac{dv}{dt}$$ are the velocities with respect to mirror not with respect to ground. When the mirror is at rest, then velocity of object or image with respect to mirror is same as velocity of object or mirror with respect to ground.