Here's a bit of a puzzle for everybody to think about. Let's set up a hypothetical situation:
You have a bow and arrow. There is a turtle at some distance away from you (say, 100m; it doesn't matter). You fire the arrow at the turtle, aimed perfectly. Now think about this:
The arrow has to pass through the half way point between you and the turle (in this case, 50m).
Then it has to pass through the halfway point between itself and the turle (in this case, 75m).
Then the halfway point between that and the turtle (87.5m).
And so on.
The problem is, no matter how many times you divide a number in half, you'll never get to 0.
Therefore, the arrow will spend an infinite amount of time trying to get half way between itself and the turtle, because there's always a half way point.
Assuming constant horizontal velocity (no wind resistance), you can determine how much time has passed between each half-way point indicated, and you'll see that you don't spend an infinite amount of time.
Assuming a total distance of 100m and a speed of the arrow at 50m/s (that's a fast arrow I believe). (The following equations are slightly off from real-life because they do not take into account gravity; if you were to factor in gravity, you would need to aim upwards, and so an initial v = 50m/s would decrease according to the vertical angle and a
g=9.8m/s
2).
At halfway point 1, you'll have spent 1s to get to where the arrow is.
At halfway point 2, you'll have spent 1.5s total to get to where the arrow is, or 0.5s from halfway point 1.
At point 3, you'll have spent 1.75s total, or 0.25s from halfway point 2.
And so on.
Your distance traversed becomes infinitesimally small as you divide by 2. However, your time travelled to reach each point also becomes infinitesimally small as you travel from one point to the next. The relationship is modeled by the function:
where d is the total distance traveled, t is time elapsed, v(t) is the velocity as a function of time, and d' is the initial distance from the origin of measurement.
Your series is simply a summation of the total distance:
This is an infinite geometric series. Solving by the property:
Your series sums to 2. I'm not exactly sure what the 2 means... but I'm sure it means something. In any case, it does indicate that the series is convergent and not infinite.
In any case, it takes d/v time to traverse a distance.
If you want to get technical, it takes t=(d-d')/v(t) time.
Congrats to MyndFyre for winning the uberl33t award for killing turtles!
