How long it will take to travel to the nearest star ?

Besides the Sun the closest start to Earth is Proxima Centauri (red dwarf) some 4.24 light years away.
Part of triple star system of Alpha Centauri A, Alpha Centauri B and Proxima Centauri.
BTW, recently scientists found Earth size planet orbiting Alpha Centauri B, no luck though it is very close to the star)
It seems like a small distance unless you start contemplating that one light year is the distance traveled by a light beam over year. Is in it fascinating that closest star to our Sun is a triple star system, not a single star as would common sense prescribe !! I hope someone has idea why that is the case ? May be it acts for our Sun in the same way Jupter acts for Earth i.e. protecting us from big space derby.
Let me go on another tangent here. Astronomers have found wanderer planets (hmm .. planet in Greek mean wanderer, should I say wanderer Wanderer :) which in the past were part of solar system, but were catapulted from their parent stars and now wander in interstellar space. They are very hard to spot as you may expect.
Ok back to the topic....
As probably you also know the speed of light is humongous compared to our everyday experience, approximately 299 792 km/s. What does this means in kilometers :

`((3 * 10^8 m/s) * (365\ days * 24\ hours * 60\ min * 60\ secs)) / (1000\ m) = 9.46 * 10^12\ km` i.e. 9.46 trillion kilometers (or 5.87 trillion miles)

Our 4.24 ly are nothing less than 40 trillion kilometers, wow !? it will be a long trip ... So light is fast, how about human ship, how fast it can go ?
The fastest ship humanity ever build the Voyager is cruising with 17 km/s at the edge of the Solar system. Apart from it there are the two Helios probes flown towards the Sun, the faster one achieved 70 km/s. (Helios spacecraft)
So we will use those two speeds to make our calculations. Here is how we will do it, we know that light takes approx 4 years to reach Proxima. We will first see how much slower our speeds are compared to the speed of light and then multiply by this ratio to get the time it will take us with that speed.
F.e. 17 km/s = 17 000 m/s, so we divide the speed of light by that number :

`(3*10^8) / (17 * 10^3) = 17647.1` times slower than the speed of light.

which means it will take :

`17647.1 * 4.24 = 74823.5\ years`

to reach Proxima Centauri.
What about if we go with 70 km/s :

`((3*10^8) / (70 * 10^3)) * 4.24 = 18171.4\ years`

So with fastest human ship we will reach there in more than 18 thousand years.
Think about it ! The first known civilization, the Summer ("land of civilized kings", "native land", ~4500 BC) were around 6500 years ago.
We are talking about three times longer, clearly such a trip is untenable. What could we do ? Go faster of course !!!
Is there something on the horizon in the near future that can help us achieve faster speeds ?

Let's turn the question the other way around..

Can we build a ship to take the trip in 100 years ?

If you are willing to understand how we will go about answering this question it is probably good idea to first read the article about the rocket equation and then continue to read below. If you are too bored to read the details just jump to the final conclusions.

OK, you decided to read :) nice of you ...
For answering this question let's find what are the current best of breed spacecraft engines. Here is what I found on Internet about top of the line engines at the moment. We are interested in the `I_(sp)` (specific impulse). Before we start calculating I should mention that we make a lot of assumptions.
First we assume that the ship is flying with constant acceleration until reaching Proxima in a straight line, second we disregard any relativistic effects, we also presume that power and propellant is 99% of the total weight of the rocket and the payload is the other 1%.
The electric engines are more efficient than chemical, but the drawback is that they require power plant and/or solar panels, which increases the weight requirements i.e. decreases the efficiency
On the plus side we can make the multistage and get rid of some of the mass over time, but this will complicate our calculations so we will disregard this too.
We are targeting 100 years trip. So we need to find : First we will use the motion equation from Classical mechanics for moving objects with constant accelerations :

`d = x_i + v_i*t + (a*t^2)/2` `d` - is the distance (in our case 4.24 light years) `x_i` - initial position (in our case 0) `v_i` - initial velocity which is also zero `a` - is the acceleration we are looking for `t` - is the time we are targeting i.e. 100 years let's derive the acceleration from this : `d = (a*t^2)/2` `a = (2*d)/t^2` `a = (2*4.24\ ly)/(100 y)^2 = (2 * 4.24 * (3 * 10^8 * 365 * 24 * 60 * 60) )/ (100 * 365 * 24 * 60 * 60)^2 = 0.00806697 = 8*10^-3 m/s^2` in other words the ship have to accelerate with `~8 (mm)/s^2`

Next thing we will use our knowledge that acceleration is velocity divided by time :

`a = (Delta v) / (Delta t)` `a * (t_f - t_i) = v_f - v_i` `a * t = v_f` `v_i` - initial velocity, which is zero `v_f` - final velocity, that's what we are looking for. `t_i` - initial time, which is zero in our case `t_f` - is `t`, the final/total fly time let's rearange the above equation : `v_f = a*t` `v_f = 8*10^-3 * 100\ y = 2.52288 * 10^7\ m/s = 8*10^-3 * 100 * (365*24*60*60) ~= 25 000\ (km)/s ` when spacecraft arrive at Proxima it would have achieved 25 000 km/s.

Finally lets see how efficient the engine has to be using the rocket equation :

The rocket equation is : `Delta v = v_e * ln\ (m_i/m_f)` where : `Delta v` - that is the famous delta-v i.e. what change of velocity will be generated. We already found out what velocity we need to achieve (25000 km/s), so that is what we are looking for. `v_e` - is the exhaust velocity of the propellant/fuel `m_i` - initial mass of the rocket (we pick 100 tonnes i.e. 100%) `m_f` - final mass of the rocket (1 tonne i.e. 1%, also payload mass) but we know that specific impulse `I_(sp) = v_e/g`, so we will substitute in the rocket equation : (`g` is earth gravitation = 9.8 m/s) `Delta v = I_(sp) * g * ln(m_i/m_f)` and if we rearrange : `I_(sp) = (Delta v) / (g*ln(m_i/m_f))` remember this whole exersice we were looking for the special impulse, so we can match it with the data we have for the current breed of engines. `I_(sp) = (2.52 * 10^7)/ (9.8 * ln(100/1)) = 558379\ s` Which btw translates to approx ~55 km/s fuel exit velocity.

So to summarize we need an engine with specific impulse around 30 times more efficient than the current available DS4G prototype to achieve a mission to Proxima that will cross the distance in 100 years.

Final remarks

So how long it will take a robot spacecraft to travel so it can reach the closest star Proxima Centauri ? Depending on three scenarios using the best of todays technologies, we still will need thousands of years to achieve this goal : From everything it seems we have to look for a new technologies and approaches.
Just as side note, Proxmia Centauri won't be always the closest star. The heavens may look eternal and stale but the stars are in a constant motion and in 30-40 000 years Ross 248 will be at a distance of ~3 ly from the Sun (near stars). And in even furthest in the future ~1.5 mln years Gliese 710 will come as close as ~1 ly, in the midst of the Oort cloud. We have to be truly space faring creatures by then to stop the probable commet bombardment of the inner Solar system.
Not seen this apocalyptic scenario in the movie yet ;).