Shining light on a retreating object

[MackBlanch]

Homework Statement

I came up with this thought experiment last night, but I'm not confident in my solution--mostly because I forgo the time values I thought would be necessary to complete it.

• A reflective object moves past a lamp at time T₀ with constant (non-relativistic) velocity V.

• Some time later, at time time T₁ the lamp turns on.

• The lamp turns off at time T₂ when struck by the light reflected back by the object.

• The lamp stops being struck by reflected light at time T₃.

Are the following durations of equal length? If not, then order them by length of duration:

1) The amount of time the lamp is 'on' (i.e. How long the lamp emits light): t_LampEmits
2) The amount of time the object is reflecting (i.e. How long the object 'sees' the lamp): t_{ObjectReflects}
3) The amount of time the lamp gets struck by reflected light. (i.e. How long the lamp 'sees' the object): t_{ReflectionsStrike}

Homework Equations

'c' is defined as the speed of light.

Light will travel a distance D in time D/c.

The Attempt at a Solution

The first light ray emitted by the lamp will strike the object and be reflected when the object is some distance, D₀, away. This first ray will then travel a total distance to and from the object of:

2 * D₀​

The final ray emitted by the lamp will travel some distance, D₁, to and from the object for a total distance of:

2 * D₁​

Because the object is moving away from the lamp,

D₀ < D₁.​

The lamp is on for the amount of time it takes light to go to and from the object,

t_LampEmits = 2 * D₀ / c​

The object reflects from the time light first strikes it at distance D₀ until light stops striking it at distance D₁. During this time, light will travel a distance, D₀, from the object to the lamp plus a distance, D₁, from the lamp to the object (with no overlap as a condition in the question). That is,

t_{ObjectReflects} = (D₀ + D₁) / c​

The lamp will be struck by light for as long as it takes its final emission to travel to the object at D₁, and back.

t_{ReflectionsStrike} = 2 * D₁ / c​

So, the durations will be different, and since D₁ > D₀, ordered as follows:

{ t_{ReflectionsStrike}, t_{ObjectReflects}, t_LampEmits }​

In prose:

The lamp will see the object for longer than it reflects, while the object will reflect for longer than the lamp is 'on'.

It seems reasonable then, that for a stationary object these durations are equal, while for an approaching object, the order is reversed.​

 

[mfb]

Correct - at least as long as we can neglect relativistic effects for the object.