Why is the preposition omitted?

Question

Examples:

1. I'm busy doing homework.

2. I could use your help getting the tire back on car.

Why there's no prepositions, like :

1. .....in doing homework.

2. .....in getting the .....

I come across more such sentences.

Please explain. Thank you.

Answer 1

English grammar does not always require prepositions in such cases. Indeed, if you included any it would seem unnatural. On the other hand, if you phrased it differently there might be prepositions.

I'm busy doing homework.
I'm busy with homework.

I could use your help getting the tire back on the car.
I could use you help with the tire change.

Basically, help doesn't require such a preposition when you use a gerund-participle to describe what the help is with. In one analysis, they are adverbials - but not all adverbials use a preposition.

Answer 2

There are no prepositions because in these sentences "doing" and "getting" are not functioning as noun. Instead they are functioning as adverbs. They modify the actions.

Contrast this with the following artificial sentences:

I am busy in the doing of my homework.

I need your help with the putting of the tire back on my car.

Here "doing" and "putting" are the names of activities. They function as nouns and so it is appropriate to make them the objects of prepositions. Note that we use articles too.