Why the following sentence is "incorrect"? "I cycled to school this morning for half an hour."

That doesn't sound *wrong* to me; it sounds *unusual*. I'd be more likely to say something more like "I biked to school this morning. Took me half an hour." or "I rode my bicycle to school this morning; it took me half an hour."(Note that the first one is informal.) I'd generally only use "I biked/ran/walked for half an hour." if I was out for exercise and didn't have a specific destination in mind (other than returning home).

since others have mostly answered your question, there is an error in your title. it should be "Why **is** the following sentence [etc...]"

I don't see anything wrong with the sentance being asked about, tho Is is oddly placed for a question

Usually when you have a prepositional phrases of time (for..."time") it sounds more natural if you put it directly after the word for whatever happened during that time. (In this case "cycled") For example: "I studied FOR two hours every day this week." Sounds much better than: "I studied every day this week for two hours"

I cycled to school this morning for half an hour and walked the rest of the way.

As noted by others implies incomplete action. You could perhaps say, for example, “I cycled to school this morning for half an hour before I had a puncture and had to stop.”

To me, saying "I cycled to school this morning for half an hour." Implies that you didn't cycle all the way to school. You traveled for half an hour then stopped when got there a different way.

I use these type of sentences a lot in casual conversations with my friends, and I tend to use them with the following emphasis: “I cycled to school this morning FOR 30 MINUTES (before realizing school was closed today)”. There is a sense of the 30 minutes being longer than expected, or being a waste of effort. “I cycled to school this morning IN (JUST) 30 MINUTES”. There is a sense of completing the action within limited time or quicker than expected. If someone asked me “how much time did you take to get to school this morning?”, i could reply with the above 2 options if I want to emphasize how quickly/slowly i got to school. My choice for a neutral response would be “It took me (around) 30 minutes to cycle to school this morning”. (note that you can also add emphasis on this sentence to imply that it was a long/wasteful 30 minutes. “It took me 30 (WHOLE) MINUTES to cycle to school this morning)

One of your examples has the wrong tense. Should be "I will be cycling to school for the next three years", or more likely, I will need to cycle to school for the next three years"

It seems like the clause "I cycled to school this morning for half an hour." standing on its own is wrong, because there is a confusion in implicature (without any further context). While the clause is grammatically correct - in "The Cambridge Grammar of the English Language - chapter 8.7.3" it is marked as incorrect. The confusion stems from the reading of the situation. Is this activity (atelic) or accomplishment (telic)? Am I on the right path? 2. Furthermore, but then why doesn't "I cycled to school this morning for half an hour" sound right to the ear? And "I cycled for half an hour to school this morning" is OK. Does the shift in position of the adjunct "for half an hour" change the focus?

To me, “I cycled for half an hour” and “I cycled to school” don’t go together, because “I cycled FOR half an hour” feels like you were going on a leisure ride and that’s how long you chose to cycle. It’s as if cycling is “mass” rather than “countable.” The trip to school is a specific distance; it’s only “countable.” I’d say “I cycled to school this morning IN half an hour.” Also, that statement implies that 30 minutes was an unusually short time. Habitual actions use present tense and wouldn’t specify “this morning,” so if that’s the time it typically takes, I’d say “It takes me half an hour to cycle to school.” If it usually takes me ten minutes and today it was 30, then I’d say “It took me half an hour to cycle to school this morning.”

I cycled to school this morning for half an hour, but then decided to turn around and go home.

"For (duration of time)" is typically used for actions without a definite end, so it's sounds quite strange when used for something like traveling to a specific location, though it can work if you structured it as: "I cycled for half an hour to get to school", however this kind of sentence would be used to emphasize that it took a considerable length of time. Example: I rode the train for 5 hours until London. I was on the plane for 8 hours before we landed in Tokyo. So for a simple sentence like this, as others have stated, it's better to use 'in' instead of 'for'. Your sentence as is sounds like you cycled *towards* school for half an hour but still didn't reach it.

It makes it seem like you didn't actually get to school, just that you cycled there for half an hour but then stopped for some reason.

It took me half an hour to cycle to school this morning. I cycled for half an hour before school this morning.

The sentence is not incorrect. It is ambiguous. You don't make it clear what 'for half an hour' relates to. I would suggest 'I cycled for half an hour this morning to get to school' is unambiguous.

Who said it was incorrect?

Perhaps this distinction can help: I \[did an activity\] for \[a period of time\]. I \[completed a finite activity\] **in** \[a period of time\]. \----------------- I ran for an hour I ran the marathon **in** 4 hours and 22 minutes. He vacuumed for 1 hour. He vacuumed the hallway **in** 13 minutes. She drove a taxi for three years. She drove to school **in** 18 minutes. We ran track for 3 years during college. We both ran 200 meters **in** less than 30 seconds. I cycled to school for 3 years. I cycled to school this morning **in** half an hour.

Other posters have already given great answers. For some further reading on this, read the Wikipedia page on telicity.