Question: I noticed that there is sentence construct that starts with an adjective, something like this:

> 
> Ready to race, John started the car.
> 
> 
> John, ready to race, started the car.
> 
> 
> 

I looked at some resources [here](http://www.chompchomp.com/terms/relativeclause.htm) and noticed that it's neither a relative clause (which should have a relative pronoun) nor an adjective phrase.

What is that structure, if I'm correct that it's indeed grammatical, called?

Answer: 
> 
> 1) *Ready to race,* John started the car.
> 
> 
> 2) John, *ready to race,* started the car.
> 
> 
> 

The expression *"ready to race"* is an **adjective phrase.** It is functioning as a **predicative adjunct,** and its predicand is the noun "John".

Consider:

* John is *[ready to race].*
* John is *[ready].*
* John is *[tall].*

The above three examples are copular clauses, where the expression in brackets is the *predicative complement*, and the predicand is the subject "John".

In your two original examples, "John" has the property "ready to race" ascribed to it.

NOTE: The terminology in this post is supposedly consistent with that used in the 2002 reference grammar by Huddleston and Pullum et al., *The Cambridge Grammar of the English Language* (CGEL). Related pages: 262-3 ([30.ii] *"[Angry at this deception,] Kim stormed out of the room"*), 529-31, 541, 545-6 ([29]).
The problem is where you find it in traditional grammars. Most likely in the chapter participle constructions. There are two possibilities to derive the above construction:

1 John was ready to race and started the car.

2 As John was ready to race he started the car.

In both cases you can shorten the first part of the two sentences by using a participle construction:

3 Being ready to race John started the car.

This is a stylistic device to give more variety to sentence structures.
If the participle construction contains "being" it is mostly dropped 
as "being" has almost no meaning or is self-evident.

Any sentence or subordinate clause can be shortened and changed into a participle construction anď added to another sentence.