Two Types of Image — Question 6

Question 6

A changing ray slope moves the reconstructed image nonlinearly

A symmetric outgoing bundle has upper ray $y=a-b(t)x$ and lower ray $y=-a+b(t)x$, with fixed positive a. The dimensionless slope parameter b is positive and differentiable.

1. Find the image coordinate and its velocity in terms of b and its time derivative.

2. If a=2 cm, b=0.10, and db/dt=0.020 per second, find the image position and velocity.

Question 6 - Solution

A changing ray slope moves the reconstructed image nonlinearly

(a) Method and calculation

$$a-bx_I=-a+bx_I\Rightarrow\boxed{x_I=a/b}.$$$$\boxed{\dot x_I=-a\dot b/b^2}.$$

(b) Interpretation and further calculation

$$\boxed{x_I=2/0.10=20\,\mathrm{cm}}.$$$$\boxed{\dot x_I=-2(0.020)/(0.10)^2=-4.0\,\mathrm{cm/s}}.$$

Physical check

Increasing convergence brings the image closer to the output plane; its motion need not equal the motion of any material object.