How does time dilation affect the perception of light speed in a moving plane?

[EgpYo]

Homework Statement

A plane is moving at a constant speed of 300m/s. If you are inside and you shine a light towards the back of the plane, how will light appear to move from the Earth's frame of reference?

Homework Equations

None

The Attempt at a Solution

Light always moves at 3x10^8 m/s so it would appear to be moving at the same speed for somebody inside the plane and somebody standing on the ground.

I just have a question about time dilation. Which person will see the event happen the fastest? Because to me it would seem that the light would hit the back of the plane fastest to somebody on the ground. This is because once the light is fired, the back of the plane is moving towards the light beam for somebody on earth.

 

[TSny]

The movement of the plane relative to the Earth does tend to make the light hit the back of the plane sooner in the Earth frame. Another factor that comes into play is length contraction. How would this influence the result?