What is the relation between image velocity, object velocity and mirror velocity? Suppositions used:
Velocity of image = VI
Velocity of object = Vo
Velocity of mirror = VM
I Know the fact that  VI=-Vo supposing mirror at rest 
and  VI=2VM supposing object at rest 
Now considering both mirror and object in motion, VI=2VM - Vo I ended up with this equation but my reference book suggests VI=2VM + Vo I am stuck on this for last 4 hours. I searched over internet and found the same expression like that of mine in a youtube video, I did not find much reference on this topic though. Tried many ways but all ended up on this simple argument, which equation to follow?
Help
 A: When in doubt draw a picture!
Let's start with the positions first and then take the derivative to end up with the velocities.

Since the image and the object are at same distance from mirror,
$$x_i-x_m=x_m-x_o \implies x_i=2x_m-x_o$$
To make sure all the signs are correct I take $x_0=95,x_m=100$ and $x_i=105$. This gives $x_i=2\cdot100-95=200-95=105$. This formula seems correct.
Taking the derivative of both sides transforms every $x$ into a velocity
$$v_i=2v_m-v_0.$$
Taking $v_m=0$ gives $v_i=-v_o$ and taking $v_i=0$ gives $v_i=2v_m$. So why did the reference come up with something different? Maybe it was a typo or maybe they defined $v_i$ to be in the opposite way. If you define $v_i$ to be the velocity towards the mirror you'll get a plus sign. You would also get $v_i=v_0$ if the mirror isn't moving. Defining it this way can has some benefits. For example if the mirror is static both velocities are positive and you don't have to worry about minus signs but personally I like this definition better (you have to think less).
A: I think that, since the velocities are in different directions, $2V_M -\left(-V_O\right) = 2V_M+V_O$.
A: yeah the proper equation is 2Vm = Vi + Vo. now you need to choose an axis and put the proper sign for the two values. the equations will give you proper magnitudes and directions for the axis that you have chosen.
A: I have considered an example and proved this formula in the image shown.

*

*The first image has plane mirror and the person at rest.

*Let us now consider the person velocity as $V_o=\frac{Unit Block(UB)}{second}$ towards the mirror and mirror velocity as $V_M=\frac{UB}{2}$ towards the person.

*The second image shows the situation after one second. We see that image velocity is $V_I=2UB=2\frac{UB}{2}+UB$.

*The third image considers the mirror velocity as $V_M=\frac{UB}{4}$ towards the person and the person velocity as $V_o=UB$ (see the third image with respect to first image). We see that the image velocity is $V_I=\frac{UB}{2}+UB=2\frac{UB}{4}+UB$.

*From 3. and 4. points, we see that the formula is $V_I=2V_M+V_o$

A: I think its for two cases
case I; when object and mirror moves along same direction Vi=2Vm -Vo
case II: when object and mirror are in motion in opposite direction Vi=2Vm + Vo
