Our honourable scholars, assalaamu alaykum, my question is about this hadith: Whoever catches a Rak'ah (unit of prayer) of a prayer before its time has expired has caught the entire prayer in its time.
Abu Hurairah reported that the Prophet, sallallaahu ʻalayhi wa sallam, said, "Whoever catches one Rak'ah of the prayer has caught the prayer." (Related by "the group") This refers to any of the prayers.
Al-Bukhaari has recorded, "Whoever of you catches one prostration of the afternoon prayer before the sun has set should complete his prayer. If one of you catches one prostration of the morning prayer before the sun has risen, he should complete his prayer."
Here, the meaning of prostration is Rak'ah. The clear meaning of the hadith is that one who catches one Rak'ah of the morning or afternoon prayer should complete the prayer even if the sun is setting or rising, although those are times in which it is disliked to pray. If one Rak'ah is performed, then the prayer is to be completed and the obligation of prayer will have been fulfilled although it is not allowed to intentionally delay those prayers until such times.
I found this in Fiqhussunnah, so I want to ask whether one Rak'at is considered complete at the first prostration or at the second? I also have heard that a Raka'h is considered complete at the Ruku' (bowing); kindly elaborate on when a Rak'ah is considered complete in order to catch it? May Allaah reward you.

All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ʻalayhi wa sallam, is His slave and Messenger. 

Scholars held different views in this regard. Some held that when the person catches up with one Rakʻah of the prayer before its time expires, he has caught up with the prayer on time. However, others maintained that when one catches up with any part of the prayer, and not necessarily a full Rakʻah, such as a prostration or even Takbeerat Al-Ihraam (the opening Takbeer), before its time expires, then he has caught up with the prayer.

Ibn Qudaamah said in Al-Mughni:

“The same applies to all the prayers; when the person catches up with one Rakʻah of the prayer before its time expires, he has caught up with the prayer. The Prophet, sallallaahu ʻalayhi wa sallam, said, ‘Whoever catches up with a Rakʻah of the prayer (before its prescribed time ends), then he has caught up with the prayer on time.’ [Al-Bukhari and Muslim] Another version of the hadeeth reads, 'Whoever catches up with one Rakʻah of the ʻAsr prayer before the sun sets has caught up with the ʻAsr prayer.' [Al-Bukhari and Muslim] I do not know of any difference in opinion among scholars in this regard.” [Al-Mughni]

He also said:

“The question arises as to whether one catches up with the prayer if he manages to perform what is less than one Rakʻah or not. There are two reported scholarly views in this regard. The first view suggests that when one performs any part of the prayer less than one Rakʻah (such as a prostration or the opening Takbeer) before the time of the prayer expires, he has not caught up with the prayer. This is the apparent indication of the statement of Al-Khiraqi and the view of Imaam Maalik given the apparent indication of the hadeeth cited above. The hadeeth clearly underlined that when the person catches up with one Rakʻah of the prayer before its time ends, he has caught up with the prayer. This indicates that when the person performs any acts of the prayer less than one Rakʻah before the time of the prayer expires, he has not caught up with the prayer in its due time, similar to the case with the Friday prayer... ”

The second view suggests that when one performs any part of the prayer and not necessarily one Rakʻah before its time expires, he has caught up with the prayer. Al-Qaadhi said, “The apparent indication of Imaam Ahmad's statement suggests that when the person manages to perform any part of the prayer before its time expires, he has caught up with the prayer.” Abu Al-Khattaab said, “Whoever manages to recite Takbeerat Al-Ihraam (opening Takbeer) before the time of the prayer expires, he has caught up with the prayer.” This is the view held by Abu Haneefah. Ash-Shaafiʻi held two scholarly views in this regard like the two views in reference. And because Abu Hurayrah reported that the Prophet, sallallaahu ʻalayhi wa sallam, said, “If any one of you catches up with a prostration of the ʻAsr prayer before the sun sets, let him complete his prayer, and if he catches up with a prostration of the Fajr prayer before the sun rises, let him complete his prayer." [Al-Bukhari and Muslim] The version of An-Nasaa’i reads, “...then he has caught up with it.” And with regard to the issue of catching up with prayers, if it has to do with one of the rulings on the prayer, then catching up one Rak’ah or less is the same, such as the case of catching up with the congregational prayer and the traveler catching up with the prayer of a resident. The first hadeeth may refer to that (the first opinion) with its Mafhoom (the implied meaning), but the Mantooq (the pronounced meaning) is given priority over Mafhoom (i.e. the first Hadeeth implies that catching up with a Rak'ah is a condition of catching up with the prayer in its fixed time, but the second Hadeeth explicitly means that when one performs any part of the prayer and not necessarily one Rakʻah before its time expires, he has caught up with the prayer). Also, the legal analogy is invalidated by catching up with one Rakʻah without the Tashahhud." [Al-Mughni]

What is meant by catching up with one Rakʻah before the time of the prayer ends means managing to perform the Rakʻah in full: reciting the Faatihah, observing the condition of Tuma'neenah (tranquility in all the positions of the prayer), standing up straight after performing the pillars of the prayer such as Rukoo’ and prostration and the like. In the Maaliki book Minah Al-Jaleel, after citing Khaleel's statement, “When the person catches up with one Rak'ah of the Fajr prayer before its time expires, he has caught up with the prayer on time,” Sheikh ʻUlaysh said:

“This means that the praying person must perform the Rakʻah in full, reciting the Faatihah moderately, observing the condition of Tuma'neenah, standing up straight after performing the Rukooʻ, and performing the Sujood (prostration). He must leave out the acts of Sunnah in this case, such as reciting another chapter of the Quran other than the Faatihah and what exceeds the Tuma'neenah in all the positions of the prayer; that is to be able to perform the Rakʻah before the time of the prayer ends...”

The Shaafiʻi scholar Zakariyya Al-Ansaari wrote:

“If the person manages to perform one Raʻkah of the prayer before its time ends and not less than one Rakʻah, then he has caught up with the prayer in its time. This is evidenced by the hadeeth cited in Saheeh Al-Bukhari and Saheeh Muslim which reads, ‘Whoever manages to perform one Rakʻah of the prayer before its time expires has caught up with the prayer in time.’ The difference between the two cases (performing one Rakʻah and performing less than one) is that the one Rakʻah includes most of the acts of the prayer in general and the rest of the prayer could be seen as repetition of these acts. This is not the case when the person performs less than one Rakʻah. He underlined that the person still bears a sin for performing part of the prayer beyond its prescribed time although it would be considered to have been performed on time...” [Asna Al-Mataalib]

Ibn Hajar (Al-Haytami) wrote, “...that is to be able to perform a complete Rakʻah concluded with the second prostration...” [Tuhfat Al-Muhtaaj]

Al-Bujayrimi wrote, “If the remaining time is not enough for him to perform all the pillars of the Rakʻah, it is impermissible for the praying person to observe the acts of the Sunnah in this Rakʻah; in this case, he must confine himself to observing the obligations only (so as to be able to complete performing them before the time of the prayer ends)...” [Haashiyat Al-Iqnaa’]

On the other hand, according to the view of the majority of scholars, in the congregational prayer, the latecomer catches up with the imaam if he performs the Rukoo’ before the imaam raises his head from Rukoo’. Ibn ʻAbd Al-Barr wrote:

“The majority of the scholars maintained that when the latecomer catches up with the imaam while performing Rukoo’ in the congregational prayer and he (latecomer) can recite the opening Takbeer properly and bows and firmly places his hands on his knees before the imaam raises his head and ends the Rukoo’, then the latecomer has caught up with this Rakʻah. The latecomer who could not catch up with the imaam in the Rukoo’ and misses the Rakʻah has also missed the two prostrations, i.e. he should follow the imaam and perform them, but they do not count for him.” [Al-Istithikaar]

Allaah knows best.