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I have begun my studies in 'Motion in a straight line', this is a really easy topic, but I don't fully understand this sign thing. I'd firstly like to point out that I understand velocity and acceleration are vector quantities and thus must have a direction associated with it. I've also read that there are different ways to show direction, however, the most common way is either a positive or negative sign before the magnitude of whatever you're calculating.

Velocity, I think, is simple enough. The sign is determined by whether the object has moved in the positive direction or the negative direction. However, deriving the sign for acceleration I find quite tricky and I would like a explanation that makes me understand what it's going.

I really don't want to be given rules, since I can't actually understand what's going by just applying rules.

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The sign of an acceleration vector indicates whether the acceleration is opposing the velocity or not, given the convention of the sign on the velocity. When they disagree, velocity is being opposed and is shrinking in magnitude over time; when they agree, velocity is being aided and is increasing in magnitude over time.

If you throw a ball up from the ground, and call that "positive" then what happens? —The ball starts moving away fast, but slows down, stops, and then starts falling, eventually falling fast when it returns to its origin. Initially the change in displacement per change in time—the velocity—is positive, because we're calling "up" positive (the choice is arbitrary but you have to be consistent). But we know the velocity is decreasing. That is, whatever acceleration is happening it is opposing the positive velocity. Eventually this makes the velocity zero, and then it turns around and starts falling. Now the change in displacement per unit of time—velocity—is negative. The same negative acceleration is being applied, so the magnitude of the velocity begins to increase and the ball falls faster.

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Formally, a negative acceleration means that the rate of change of the velocity is negative. To visualize this concept, you can think about the forces involved. The sign of the acceleration is the same as the sign of the force needed to produce the acceleration. For example, if an object is speeding up in the negative direction, this means there is a net force in the negative direction producing this acceleration. Likewise, if an object is moving in the positive direction and slowing down, there is a net force in the negative direction producing this acceleration. Thus, in both of these cases the acceleration is negative.

You can apply an analogous argument to determine when acceleration is positive.

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