Plane Mirrors — Question 2

Question 2

Image velocity combines mirror and object motion with different weights

A plane mirror translates normally with coordinate xM(t), and an object on its reflecting side has coordinate xO(t). Positive x is fixed in the laboratory. At an instant, mirror velocity is +0.30 m/s and object velocity is -0.20 m/s.

1. Derive the laboratory image velocity.

2. Calculate the image velocity and the rate of change of image-object separation, assuming the image remains at larger x than the object.

Question 2 - Solution

Image velocity combines mirror and object motion with different weights

(a) Method and calculation

$$x_I=2x_M-x_O\Rightarrow\boxed{v_I=2v_M-v_O}.$$$$\boxed{v_I=2(0.30)-(-0.20)=+0.80\,\mathrm{m/s}}.$$

(b) Interpretation and further calculation

$$d=x_I-x_O\Rightarrow\dot d=v_I-v_O=2(v_M-v_O).$$$$\boxed{\dot d=1.00\,\mathrm{m/s}}.$$

Physical check

Image speed is a geometrical rate and can differ from both object speed and mirror speed.