Question

The ratio of the speed of an object to the speed of its real image of magnification m in the case of a convex mirror

Answer 1

We know mirror formula,
1/v + 1/u = 1/f
if I assumed mirror doesn't change , then focal length will be constant.
now, differentiate with respect to time ,
-1/v² dv/dt - 1/u² du/dt = 0
⇒u² dv/dt = -v² du/dt
⇒dv/dt = -(v/u)² du/dt

∵ magnification , m = v/u
∴ dv/dt = -m² du/dt
⇒Vim = -m² Vom
Here, Vim = velocity of image with respect to mirror
Vom = velocity of object with respect to mirror.

Now, come to the point,
given, mirror is convex mirror,
And we have to find out ratio of speed of object to the speed of its real image .
But we know, practically real image can't formed we with help of convex mirror.
theorically, ratio of speed of object to the speed of image = -1/m² , where m is magnification .

Answer 2

Convex mirror is a curved mirror for which the reflective surface bulges out towards the light source. Convex mirrors reflect light outwards (diverging light rays) and therefore they are not used to focus light.

Explanation:

We know mirror formula

$\frac{1}{f} = \frac{1}{u} +\frac{1}{v}$

Now wehave to differentiate it with respect to time

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

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

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

∵ magnification , m = $\frac{v}{u}$

∴ $\frac{dv}{dt} =-m^{2} Vom$

Here, $Vim$ = velocity of image with respect to mirror

V$v_om$ = velocity of object with respect to mirror.

As we consider convex mirror

And we have to find out ratio of speed of object to the speed of its real image .

But we know, practically real image can't formed we with help of convex mirror.

theoretically, ratio of speed of object to the speed of image =    -$\frac{1}{v^{2} }$ , where m is magnification .