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What is the rule determining whether one must use the (definite) article in the expressions like:

  1. the ball of (the) radius 1
  2. a drop with (the) charge 2q

I meet quite often both variants, but see no pattern:

  1. [...] moves with the speed equal to the velocity of water
  2. The train would reach its destination with speed equal to zero
  3. Kant's point, therefore, is that instantaneous states of motion (whether with speed equal to zero or any other value) [...]
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Here are two rules I found:

  1. The word "the" is one of the most common words in English. It is our only definite article. Nouns in English are preceded by the definite article when the speaker believes that the listener already knows what he is referring to. The speaker may believe this for many different reasons, some of which are listed below. [1.]
  2. The is called a definite article. "Definite" means "specific". Use the when talking about something which is already known to the listener or which has been previously mentioned, introduced, or discussed. [2.]

I will number your examples:

  1. the ball of (the) radius 1
  2. a drop with (the) charge 2q
  3. [...] moves with the speed equal to the velocity of water
  4. The train would reach its destination with speed equal to zero.
  5. Kant's point, therefore, is that instantaneous states of motion (whether with speed equal to zero or any other value) [...]

For 1., the ball is ok because I suppose that this ball was introduced earlier in the text. I study math, so to me it comes off as some kind of exercise in math or physics. I can imagine that the paragraph begins something like

Suppose a ball is thrown in the air. The ball of radius 1 travels at a rate of ...

Notice that I omit the from (the radius) because the radius was not previously provided. In other words, you cannot use the word the to refer to a radius because a radius was not previously defined.

For 2., again, the charge was not previously defined. So you cannot refer it it. Therefore, you omit the and write

A drop with charge 2q.

For 3., "with the speed of" sounds strange because you have not defined or provided which speed. You are about to define it as "equal to the velocity of water". I assume that you understand or previously defined what the "velocity of water" is. Because you already know what "velocity of water" is, you can use the word the.

For 4., I assume that this "train" was introduced earlier in the conversation or text, therefore it is understood what the means. Like maybe it was a blue train, or a local train, or a big train, etc.

For 5., seems fine. No need for the.

Also note, there might be some exceptions to these rules. But I could not think of any at the moment.

  • In terms of #4, I think the OP's question revolves around the phrase "with speed equal to". Can we say with a speed of 2Km/h... or with the speed of sound? – Cardinal Jun 13 '16 at 13:28
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    I see your point. Do not delete your comment. I will let OP ask if he/she needs clarification on number 4. – Em. Jun 13 '16 at 13:35
  • I think your logic is clear: you will probably say that no article is needed. – Serguei Jun 13 '16 at 16:00
  • @Serguei Yes, in the case "with speed equal to", no article is need because you have not introduced which speed. You are about to do that. In the case of "with the speed of sound", I believe there is only one speed of sound, so it is clear to what the word the refers. – Em. Jun 13 '16 at 16:03
  • Just to be sure. Do you mean, I normally must write "with speed equal to 300m/s", but "with the speed of sound"? – Serguei Jun 13 '16 at 16:57

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