On a physics assignment, I believe that these sentences are grammatically incorrect, but some other students disagree (especially on the second one).
What is the maximum speed that the mass can be whirled without breaking the string?
What is the maximum speed that this car can round this curve without skidding
(NOTE: 33 1/3 is the frequency that it turns — 33 1/3 revolutions per minute)
I think the use of that is incorrect, and "at which" or "that … at" must be used to be grammatically correct. "that …" is used above as restrictive clauses.
They accepted that the last one is incorrect.
the frequency [that it turns __ ]
Something cannot turn the "maximum speed".
For the second part, it doesn't seem like proper use of restrictive clauses.
the maximum speed [that this car can round _? this curve _?]
-> this car can round __ this curve
-> this car can round this curve __
"the maximum speed" is not the object of the clause. The car cannot round "the maximum speed".
Using "at which" or "that … at" is better:
the maximum speed [at __ which this car can round this curve]
the maximum speed [that this car can round this curve at __ ]
Compare this to some valid examples:
Subject: This is a sentence [that I wrote __ ].
Object: This is a sentence [that __ was written by me ].
Preposition: This is a sentence [at __ which you looked ].
Detached preposition: This is a sentence [that you looked at __ ].
Am I correct?
Best Answer
Grammar Girl has a rather lengthy discussion of the use of 'where' vs 'in which'. Although I realize that the OP is asking about 'that' vs 'at which', I believe it's the same basic conundrum, and that the same conclusions can apply. Summarizing some of the points in the article:
So I think any of your three examples are fine as-is, and would be equally fine using "at which" or "that...at" instead of "that". The choice should depend on your writing style and intended audience.
Edit to add per the discussion in the comments "The frequency that the car turns" is a grammatical ellipsis of "The frequency that the car turns at." (the third bullet scenario I listed.) Ellipses are incomplete, but that does not equate to incorrect by any standard I can find. Even the great writers use them:
quotation sources