What would be wrong with dropping a payload of acid to eat down a little into the rock and follow with an explosive mix?
Getting it there would be one issue. Every kg of payload would require additional fuel etc.
Second, to drop things on to it would suggest being in orbit around it... which requires slowing the craft down from the c.6.6 km/s it was travelling. That would take quite a bit of fuel, which would need to be launched from earth and carried most of the way - up to the point the craft wants to slow down. Again, that increases the weight launch quite considerably.
That's just the issue with getting it to the point of dropping the payloads. It's an engineering challenge rather than an impossibility, of course, but that brings home the next question:

WHY, for the love of God, WHY??????


NASA are also confident that the asteroid and moon posed no real threat to Earth before the impact by DART, and will pose no threat after... but time will tell! :D
Chances are that it will likely have just altered the oribit of the moon around the asteroid very slightly. I think they're suggesting it will have shortened Dimorphos' orbit by 1%.

Bear in mind that even the smaller of the asteroids, Dimorphos, that DART impacted on, is some 10 million times more massive than DART. Yes, DART was travelling faster than a bat out of hell (assuming such creatures can't reach speeds of 6.6km/s), but the mass differential is like a 1/100th of a gram compared to someone weighing 100kg.

But, maybe... just maybe... if Dimorphos does split open, we may finally uncover where Trump has hidden all the other Classified material he supposedly took! ;)
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Again take dropping a payload of acid as being launch a payload of acid at Dimorphos from Didymos
Still no.
No matter how you cut it, the potential energy in an acid is far, far short of the potential energy in an explosive. Which means pound for pound, you might as well just explode it.

And the kinetic energy of a bus moving at 15,000mph is greater than the energy you could get out of the small explosive you could put on there and still deploy it reliably.

DART massed 600kg and imparted about 1 trillion Joules of energy to its target.

A one kT bomb could produce 4 trillion Joules, but it masses 4,000kg.
And that doesn't include an engine.
And less than 50% of the explosive energy is directed to the target.

The numbers just don't add up, no matter how many "tweaks" you apply.
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