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During my driving lesson, we went from a side road onto the highway. This relates to physics because the horizontal acceleration of the car changed quickly. However, the vertical acceleration stayed the same because of gravity. Gravity on Earth is always 9.81 meters per second squared down.

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True, but you weren't accelerating downward.  Your weight (the force of gravity on your car) was mg, where g was 9.81 m/s2, but you weren't accelerating vertically.  (PS -- not quite a blog post for the class -- see me to talk about where to put them).

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