# The Time Machine Invented

**Classical Mechanics**Level 3

**Theorem:**Any vehicle which can travel at infinite speed can also travel (backwards or forwards) in time.

**Corollary:** Any vehicle which can travel at a reasonably high speed can also travel (backwards or forwards) in time.

**Solve the problem given after the proof.**

**Proof of theorem:** Consider cities A, B, C, D… located in consecutive 1-hour time zones around the world.

Suppose you are at city A at 7 am on 7th June. You get into your vehicle and instantaneously travel to city B at infinite velocity, and arrive there instantaneously. Now you are 1 hour in the future, and you have to set your clock to 8 am. Travelling further, at city C, you are at 9 am, at city D at 10 am and so on. At city R, it is midnight, and sure enough, at city S, it is 1 am the next day, 8th June.

So if you instantaneously travel around the world at infinite speed from west to east and come back to city A, you are 1 whole day into the future, i.e. it is 7 am on 8 June instead of 7 June. If you make N trips around the world at infinite speed (i.e. instantaneously), from west to east, then you are N days in the future.

If you made the trips from east to west, i.e. from city D to C to B to A, you would set your clock 1 hour back at each city, i.e. 7 am, 6 am, 5 am…midnight previous date, … until you arrive back at city A 1 day in the past. If you make N trips around the world instantaneously from east to west, you would be N days in the past.

In order to convince yourself that the above is actually true, consider that there actually is something called an International Date Line, where the date actually changes. When you cross over the date line, you are one day in the past or future, depending on the direction in which you cross it. So if you go around the world instantaneously and cross the international date line N times in the same direction, then you are actually N days in the past or future.

Superman actually did this in one of his movies, and this proof shows that that was not fiction, but a scientific fact.

**Proof of Corollary:**

Suppose you have a vehicle which does not have infinite speed, but is still very fast, and you can travel around the world in 5 minutes. Now, starting at 7 am on 7 June and going from west to east, around the world in 5 minutes and coming back to city A, the time would be 7 am on 8 June, so I go from 7 am, 7 June to 7 am, 8 June in 5 minutes. So, other than the 5 minutes which I have spent, I have traveled 23 hrs, 55 minutes into the future. With each of N rounds around the world at this speed, I travel 23 hrs 55 minutes further into the future.

Now suppose, I have a still slower vehicle, which travels around the world in 23 hrs, 55 minutes. Starting at 7 am on 7 June at city A, and traveling west to east, it would take me 23 hrs, 55 min to reach city A again, where the time would be 7 am, 8 June, so other than the 23 hrs 55 min which I have spent, I have traveled 5 minutes into the future. Travelling around the world at this speed, with each revolution, I travel 5 minutes into the future, so with N revolutions, I am 5N minutes in the future, where N can be as large as you want. With N revolutions from east to west, I can be 5 N minutes in the past, where N can be as large as you want.

This phenomenon is actually experienced by millions of travelers in daily life. If a traveler flies from London to New York, he has to set his clock 5 hours back (say from 10 am to 5 am), so he goes back to the past, and re-lives the same 5 hours of his life which he had already lived before. He could keep flying westward, keep going to the past, and re-living the same hours again and again, any number of times. Similarly, a traveler who flies eastward, goes to the future (say from 5 am to 10 am) and continues his life from that point in the future, without ever having lived the intervening hours.

**Consider the above proofs carefully and choose one option.**