Vtyjkyuk

"Hard" is sometimes used as an adverb in English to emphasize a physical action, or indicate that it was especially vigorous or forceful. For example, "he hit the ground hard when he fell", or "she punched the tree so hard she left a mark".

How would I express this idea most clearly in Latin?

As a general expression that's applicable to a wide variety of actions, graviter would work.

In Caesar, De bello gallico 3.14.4, it's used (in the comparative degree) of falling spears:

rostro enim noceri non posse cognoverant; turribus autem excitatis tamen has altitudo puppium ex barbaris navibus superabat, ut neque ex inferiore loco satis commode tela adigi possent et missa a Gallis gravius acciderent.

for they knew that damage could not be done by their beaks; and that, although turrets were built [on their decks], yet the height of the stems of the barbarian ships exceeded these; so that weapons could not be cast up from [our] lower position with sufficient effect, and those cast by the Gauls fell the more forcibly upon us.

(Translation by W. A. McDevitte and W. S. Bohn, from the Perseus website)

In Suetonius's life of Otho 7.2, it's used of taking a tumble:

dicitur ea nocte per quietem pauefactus gemitus maximos edidisse repertusque a concursantibus humi ante lectum iacens per omnia piaculorum genera Manes Galbae, a quo deturbari expellique se uiderat, propitiare temptasse; postridie quoque in augurando tempestate orta grauiter prolapsum identidem obmurmurasse:

τί γάρ μοι καὶ μακροῖς αὐλοῖς;

He is said to have been greatly frightened that night in his sleep, and to have groaned heavily; and being found, by those who came running in to see what the matter was, lying upon the floor before his bed, he endeavoured by every kind of atonement to appease the ghost of Galba, by which he had found himself violently tumbled out of bed. The next day, as he was taking the omens, a great storm arising, and sustaining a grievous fall, he muttered to himself from time to time:

"τί γαρ́ μοι καὶ μακροῖς αὐλοῖς;"

"What business have I the loud trumpets to sound?"

(Translation from the Perseus website)

And in Pliny the Younger, Epistulae 3.14.7 (one of my favorite letters, by the way), it's used to describe an episode in which a Roman knight strikes and nearly kills a man who will later be killed by some of his own slaves:

eques Romanus a seruo eius, ut transitum daret, manu leuiter admonitus conuertit se nec seruum, a quo erat tactus, sed ipsum Macedonem tam grauiter palma percussit ut paene concideret.

One of Macedo's slaves lightly touched a Roman knight to ask him to let them pass; the man turned round and struck not the slave who had touched him, but [gave] Macedo himself such a violent slap that he nearly knocked him down.

(Loeb translation by Betty Radice)

Elsewhere, just going by the examples in OLD, graviter is used also of clashing warships, blowing winds, sounds, footfalls, etc.

Still, depending on the specific action that's involved, there are likely to be other options, possibly more evocative. For example, in the context of punching, 'so hard' could be expressed by tam vehementer or tanta vi.