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I understand what negative velocity means, but how does negative velocity and negative acceleration works together. Although it depends on frame of reference, how negative acceleration with negative velocity make it more negative. (I understand the math but want an intuitive explanation.)

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If velocity and acceleration have the same sign, you speed up.

If velocity and acceleration have opposite signs, you slow down.

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Negative velocity = You move backwards.

Negative acceleration = You slow down or you go faster in the backwards direction.

Let's say you have this simple coordinate system:

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where the arrow shows in positive x direction. If you would stand on this line and move to the right, we say that you have "positive" velocity. Vice versa, if you move left, you would have "negative" velocity. While moving to the right, you would accelerate (the acceleration would be positive, too). To slow down, you would need negative acceleration (you brake basically). And if you stand still and then you decide to accelerate negatively, you would move to the left.

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Let us assume our coordinate system goes from left (negative $ x$ ) to right (positive $x$). Positive velocity is then speed towards right, and negative velocity is speed towards left. It is the same for acceleration: negative acceleration is acceleration towards left. It slows objects moving towards right (with positive velocity) down, until they stop and move towards left, and it makes object already moving towards left (with negative velocity) move faster towards left.

Similar reasoning obviously apply for other coordinate choices ( up-down, right- left , etc.) There is nothing specific distinguishing positive and negative acceleration, the sign is just an arbitrary coordinate choice.

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