I know that a clock placed on an aeroplane will have slowed with respect to a clock placed on earth because the more our velocities are, compared to the speed of light, the more the time slows down. But, the earth is moving faster than the aeroplane when viewed from outer space. Then, the clock placed on Earth should be slower than a clock placed on the plane.
But the clocks are the same. How would that be different?
If you say that earth's velocity around the sun is 67,000 mi/h, your reference point is the sun itself, which makes the aeroplane's velocity 68,000 mi/h, not 1000.
Using special relativity only, and (A) observing from the sun, a clock on the plane would seem to run slower than a clock on earth. A person (B) on earth would measure also measure an aeroplane's clock to be slower, but by a different factor.
time dilation, $ \Delta t' = \frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}} $
(A) observation from the sun: time dilation of aeroplane clock vs earth clock, (i)$\frac{\Delta t}{\sqrt{1-\frac{68000^2}{c^2}}}$ and (ii)$\frac{\Delta t}{\sqrt{1-\frac{67000^2}{c^2}}}$
(B) observation from the earth: time dilation of aeroplane clock vs earth clock, (iii) $\frac{\Delta t}{\sqrt{1-\frac{1000^2}{c^2}}}$
These are all different values due to different velocities as measured from different observation locations. This is what relativity is all about. (side note, this equations are correct iff $c$ is in mi/h.)
Up to this point we have ignored general relativity, which takes into account time dilation due to gravitational acceleration (clocks are measured to be faster in lower g). This is an opposite effect from the time dilation due to SR. As it turns out, aeroplanes are travelling way too slow to have their clocks observed to be slowed at all--in fact they measure to be faster than earth clocks, because the difference in gravity outcompetes the difference in velocity in this case. If you're wondering if there is a sweet spot where time dilation due to SR and GR cancel out, there is. You can find out more by searching for time dilation due to gravitation and motion.
An important thing to note is that the effect of time dilation are observational effects, and are due different conditions at the observation point and the point being observed. When two objects have relative velocity, they both measure the other's clock to be slower than their own but a third object with the same velocity as one of the original two will clearly not measure those two as equally slow.
The aeroplane is moving in the atmosphere of the earth, and so how can you say that the earth is moving faster, when viewed from outerspace. I think the aeroplane will still be faster.
