Are the planets accelerating? why or why not

Traveling in any path besides a straight line at a constant velocity requires some type of acceleration.   Therefore, the Earth is always accelerating as it bends its way around the sun.  This is because the Sun is always exerting a gravitational force on the Earth.  This may seem a little counterintuitive because if you're going in a circle, you always have the same speed, but the point is that the direction is constantly changing.  In particular at opposite sides of the an orbit, the Earth's velocity have opposite directions.

The Earth has a slightly eccentric orbit, meaning that it isn't going around the Sun in a perfect circle.  If we were going in a perfect circle, the speed of the Earth would be constant. but due to the eccentricity of the orbit, we're going slightly faster when we are closer to the Sun (in January) than we are when we are further away (in July).  So there is a slight change in speed.  But this change in speed is only a small part of the acceleration.  

Weight is a slightly different story.  Weight is the upward force that the surface of the Earth exerts upon a body to keep it from falling.  The tricky thing about this question is that both the body and the Earth are falling towards the Sun with the same acceleration.  This means that the force that the surface of the Earth exerts upon the body isn't changing as they go around the Sun.

There are some residual effects that are very small having to do with the fact that one side of the Earth is closer to the Sun than the other side.  These so-called tidal forces are very small, but cause solar tides (which are smaller than lunar tides)

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