Mettra wrote:All you've done is obfuscated the problem. From the frame of reference of the earth, it would appear that Mars is accelerating harder, but that's now a non-inertial frame of reference since earth is being accelerated noticeably towards Mars. From an inertial frame of reference, Mars will experience the same acceleration as the bowling ball.
Actually, it should work in an inertial frame as well. Take an inertial frame at the position and velocity of the earth at the start of the experiment. Say that the Earth's pull is 1000 units/second2
at a distance of 100 units. And Mar's pull is 100 units/second2
at a distance of 100 units. And the bowling ball's gravitational pull is negligible. Start the objects 1000 units away from Earth, in two separate trials. I'm just going to use some crude discrete calculations for demonstration. Since the gravitational pull of the earth at 100 units away is 1000 units/second2
, the pull at 1000 units away is 10 units/second2
(gravity inversely proportional to the square of the distance). So after the first second, both the bowling ball and Mars are moving toward Earth at 10 units/second, and let's call that 10 units closer as well. But in the Mars drop trial, the Earth moves as well, the pull on it is 1 unit/second2
at a distance of 1000 units, and so it's at 1 unit/second, and say one unit closer to Mars.
Now here's where the situation changes. The bowling ball is 990 units away from Earth, but Mars is 989 units away from Earth. That means the gravitational acceleration for the bowling ball at that distance is now 10.2030405 units/second2
, whereas Mars, being closer, has a gravitational pull at it's distance of 10.223684 units/second2
. So the bowling ball in the next second, has a speed of 20.2030405 units/second from our inertial reference point, and Mars has a speed of 20.223684 units/second from our initial reference point. Earth would be up to 2.0223684 units/s for the Mars trial, and stationary still for the bowling ball. So Mars would then be at a distance of 966.7539 units, whereas the bowling ball would be at a distance of 969.7969 units. So then the gravitational acceleration of the Earth on the bowling ball at that distance would be 10.63257 units/second2
. And the acceleration of Mars, being closer, would be 10.6996 units/second2
And so on and so forth if I got the math roughly close. And so Mars, even from an inertial reference point, would be seen to have a greater acceleration over the whole trip, as can be seen looking at the speeds of both throughout the trip or the derivative thereof, because it's instant acceleration is increasing faster then the bowling ball, since it's getting closer sooner, and closer = more acceleration.
Note that this only works if you drop the bowling ball and Mars on two separate trials, since if you dropped them at the same time, the bowling ball would also be affected by Mars drawing Earth closer.