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Can I say,

(a) He accidentally kicked the ball too hard until it rolled into the drain. He ran quickly to pick it up / to pick up the ball.

(b) He kicked the ball very hard. It bounced and fell into the drain. He rain to pick up the ball.
Comments  
Corrected versions:

(a) He accidentally kicked the ball too hard until it rolled into the drain. He ran quickly to pick it up. [Its understood that the ball is the subject and repeating the word "ball" is unnecessary]

(b) Anyways acceptable [He kicked the ball very hard. It bounced and fell into the drain. He ran to pick up the ball]
Vincent TeoCan I say,

(a) He accidentally kicked the ball too hard until it rolled into the drain. He ran quickly to pick it up / to pick up the ball. The second sentence is fine in both versions. The first won't do, unless you really want to suggest that he repeatedly kicked the ball too hard accidently - This would be so odd that I think you should say He accidentally kicked the ball too hard and it rolled into the drain.

(b) He kicked the ball very hard. It bounced and fell into the drain. He ran to pick up the ball. - this is correct but pedestrian - why if you could use it to speak of the ball in the second sentence don't you use it in the third also? It would be much more natural to say He ran to pick it up.