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One of the cases in which one can perform tayammum instead of wudhu or ghusl is when there isn't enough time for the two (meaning that if one does so, his prayer will be missed and not done on time).
1- Whenever time is so short that if one performs wudhu or ghusl, his prayer will partially or fully be performed after its time, he/she must perform tayammum instead.
2- If one purposely delays the prayer until not enough time remains for performing wudhu or ghusl, he/she has sinned, nevertheless, (tayammum must still be performed and) the prayer he/she prays with tayammum is valid, although it preferred that he/she makes up the prayer as well.
3- If one isn't sure if he/she performs wudhu or ghusl there will be sufficient time for the prayer or not must perform tayammum (and once again, if he/she performs wudhu or ghusl, it will be valid, although he/she has sinned).
Therefore, in this case, tayammum is acceptable, and the prayer doesn’t need to be made up for, nevertheless, the tayammum is no longer acceptable for the ensuing prayers if there is water to be found for wudhu or ghusl and there is time. Of course, according to Imam Khomeini (rah), if one purposely delays the prayer until there isn't enough time, he/she should make up for the prayer according to mustahabb precaution (but doesn’t have to, it’s just better). Also, it should be noted that if one needs to perform ghusl for jinabah (seminal discharge) before the fajr adhan during the month of Ramadhan, and performs tayammum instead because of lack of time, it will only be sufficient for the validity of the fast of that day, but the person must perform ghusl for the validity of his/her fajr prayer (because the ghusl needs to be made before the fajr adhan in order for the fast to be valid, but when the adhan is said, there is a lot of time left for the one fasting to pray, therefore, he/she needs to perform the ghusl for the prayer).
 Ayatullah Bahjat: if time is short to the extent that all of the prayer will be prayed after the end of its time, he/she must perform tayammum, and in the case of at least one rak’ah of the prayer being done before the end of its time and the rest after it if wudhu/ghusl is done, he/she can choose between performing tayammum and doing all of the prayer before the time ends, and performing wudhu and getting at least one rak’ah done before time runs out. Nevertheless, performing tayammum and finishing the prayer completely before time runs out complies with mustahabb precaution. Now, if he/she can't fit the whole prayer in its time even if tayammum is performed, then he/she must perform wudhu and pray at least one rak’ah before time ends.
 Tawdhihul-Masa’el (Imam Khomeini’s annotations), vol. 1, pp. 377-378, issues 678-680.
 Imam Khomeini, Tahrirul-Wasilah, vol. 1, pg. 283.