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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?

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........agreed. –  Edwin Ashworth Apr 26 '13 at 21:47
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Do you know of any style guides or authorities that comment on this usage issue? As written, this sounds less like a question and more like an argument. Most likely the people who accept this usage are simply inferring at. –  Bradd Szonye Apr 26 '13 at 21:50
    
@BraddSzonye, this is a question because I want to know if I or they are correct. –  Victor Apr 26 '13 at 21:54
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Consider reducing the bias in your question (e.g., calling alternatives "obviously incorrect" versus "valid"). Also, you'll probably get better feedback and fewer downvotes if you do some research of your own. When it comes to style and acceptable usage, reasoning things through only goes so far. –  Bradd Szonye Apr 26 '13 at 22:39
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The phenomenon is called Pied-Piping, and consists of moving a whole prepositional phrase to the front of a relative clause, instead of just moving the relative pronoun. It's been extensively discussed here. –  John Lawler Apr 30 '13 at 13:44

3 Answers 3

It will probably seem less open to debate if the sentences are truncated:

What is the maximum speed that the mass can be whirled?

What is the maximum speed that this car can round this curve?

33 1/3 is the frequency that it turns.

I don't like any of them. And I wouldn't like the job of finding style guides dealing with this issue - perhaps Bradd could help you out here.

There are idiomatic examples of similar sentences without the preposition, as you say:

What is the maximum speed that this car can do / manage / attain.

But (a steady) speed / x mph are allowable objects of do / manage / attain / maintain ..., whereas they are not allowable objects of whirl / round / turn.

Most likely the people who accept this usage are simply inferring at. Ungrammatically. There is a tendency to drop the at from very common examples of this construction (What is the highest speed you are allowed to drive?), but that doesn't license the elision generally (Who do you think you're looking?)

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Look at isn't comparable because it's a phrasal verb. You can't elide at because it changes the meaning. I haven't looked at style guides yet – I was encouraging the questioner to do his homework rather than asking people to disprove (what I see as) an overly prescriptivist assertion about grammar with no evidence other than some very one-sided reasoning. –  Bradd Szonye Apr 27 '13 at 1:14
    
Truncating even more makes this easier to dislike: "What is the speed that the mass can be whirled?" To switch its topics: "What is the height that it will bump its head?" –  MrHen Apr 29 '13 at 19:05

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:

  • 'In which/at which' sounds formal, and would likely not be used in casual conversation. (My own editorializing: Some might even consider it stilted or pretentious.)
  • 'where/that' is more casual, but loses some nuance in some situations, like 'the house in which I saw you' vs 'the house where I saw you'.
  • Using 'in/at' at the end of the sentence can be considered by some to be unacceptably informal or even (in some style guides) incorrect.

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

"When well used, ellipsis can create a bond of sorts between the writer and the reader. The writer is saying, in effect, I needn't spell everything out for you; I know you'll understand." (Martha Kolln, Rhetorical Grammar, 5th ed. Pearson, 2007)

"Some people go to priests; others to poetry; I to my friends." (Virginia Woolf)

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Using where instead of in which doesn't cause an invalid relative clause, but using that instead of at which does. "The house [in ______ which I saw you]" or "the house [where I saw you ______ (place)]". However, for the above examples, if "that" is used, can you "whirl" or "round" a speed? Can you "turn" a frequency? –  Victor Apr 29 '13 at 14:39
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@Victor - I don't understand your use of ____ in your examples there. But I find no fault with "the frequency that it turns", one of your earlier examples, in place of "the frequency at which it turns". I don't see why you think it's "invalid". –  Lynn Apr 29 '13 at 14:51
    
The ______ shows if the antecedent is the subject or object of the relative clause. In "the frequency [that it turns ______]", the clause modifies "the frequency" and says that "it turns [the frequency]", which is wrong. It turns at the frequency. –  Victor Apr 29 '13 at 21:58
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@Victor - Yes, of course it turns at the frequency. But ellipsis (definition 1a) allows you to leave off the 'at' without adversely changing the meaning. It is not as verbose as it could be, but it is not wrong, either. –  Lynn Apr 30 '13 at 3:42
    
1a says "the omission of one or more words that are obviously understood but that must be supplied to make a construction grammatically complete." Notice "must be supplied to make a construction grammatically complete." –  Victor Apr 30 '13 at 23:08

The above are correct. You only have to answer the questions to see that they are defining relative clauses. The maximum speed that this car can round this curve without skidding is X. That this car can round this curve without skidding restricts the noun phrase maximum speed, otherwise, what do you mean by maximum speed: the speed of light or faster? The maximum speed that we are allowed to drive in the UK is 70mph. Again, I have restricted the noun phrase maximum speed to that we are allowed to drive in the UK. Which this car can round this curve at or that this car can round this curve at mean the same. The maximum speed is a noun phrase and is the subject of the sentence. Unfortunately, I don't understand what your further explanations mean.

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That this car can round this curve without skidding does not have "the maximum speed" as its subject or object. –  Victor Apr 26 '13 at 23:11
    
However, the maximum speed that we are allowed to drive in the UK has "the maximum speed" as the object of the clause. One can rearrange it as "we are allowed to drive [the maximum speed] in the UK" to confirm it, if "driving speeds" is allowed. –  Victor Apr 26 '13 at 23:18

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