Suppose I have a video that plays for 60 minutes at normal "1x" speed. I know that if I set the video to play at "2x" speed, then it should play for 30 minutes.
Now, what if the video is set to play at "1.5x" speed? Intuition leads me to two answers:
Since 1.5x is directly in the middle between 1x and 2x, then the video should play directly in the middle of 60 minutes and 30 minutes, meaning it should play for 45 minutes.
When playing at 2x speed, I can compute that
$$ 60\ \text{real-time minutes} \times \frac{1\ \text{play-back minute}}{2\ \text{real-time minutes}} = 30 \ \text{play-back minutes} $$Analogously, for 1.5 speed, I can compute $60 \times \frac{1}{1.5} =$ 40.0 minutes.
Which one is the correct answer, 45 minutes or 40 minutes, and why? What if I try to play the video at $Nx$ speed, where $N > 0$, to indicate playing media at a faster speed or slower speed?
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$\begingroup$Perhaps it helps to change the scenario. Suppose a tortoise had to run a $60$ metre race. At normal speed, it does one metre per minute, and so the tortoise takes $60$ minutes. At $1.5\times$ speed, the tortoise travels $1.5$ metres per minute, and so after $40$ minutes the tortoise has travelled $1.5 \times 40 = 60$ metres.
The reason that you can't simply take the mean of $30$ and $60$ is actually quite subtle. If you rearrange the formula$$ \text{speed}=\frac{\text{distance}}{\text{time}} \, , $$you get$$ \text{time}=\frac{\text{distance}}{\text{speed}} \, . $$Here, the distance is $60$. Hence,$$ \text{time} = \frac{60}{\text{speed}} \, . $$If you plot this on a graph, you get this:
At first, the time taken is decreasing rapidly as your speed increases. However, the steepness of the curve then starts to become less apparent. This means that between $1\times$ and $1.5 \times$ speed, the change in time is larger than the change in time between $1.5 \times$ speed and $2 \times$ speed. Hence, the time goes all the way down to $40$ as you go from $1\times$ to $1.5\times$ speed, which is less than $45$. Most of the 'going down' has already occured when you get to the midpoint of $1$ and $2$, if that makes sense.
$\endgroup$ $\begingroup$The second. Your first idea obtains the completion time from the playback rate given by a harmonic mean of $1\times,\,2\times$, viz. $\frac{2}{1+\tfrac12}=\tfrac43$. Similarly, $\tfrac{2}{\tfrac{1}{30}+\tfrac{1}{60}}=40$ is a harmonic mean of two completion times, obtained with the arithmetic-mean playback rate. Finally, a geometric-mean playback rate obtains a geometric-mean completion time.
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