[Physics] Why is the sign of the tangential component of acceleration negative in this problem?

accelerationkinematicsvelocity

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Since the train is slowing down at a constant velocity, shouldn't the acceleration equal zero? Why is it negative in this case?

Best Answer

The problem doesn't say the train "is slowing down at a constant velocity." That doesn't even make sense --- either it's slowing down, and the velocity is not constant, or it's not slowing down, and the velocity is constant. It can't be both accelerating and at a constant velocity, since the very definition of acceleration is change in velocity with respect to time.

Imagine you are driving in your car, dead north, which we'll call the positive direction. If you slam on the brakes, which direction will the acceleration be in? Since the car is slowing down, your acceleration will be in the South/negative direction. The brakes are decreasing the car's velocity, which is the same thing as increasing it in the negative direction. Thus the sign of acceleration is negative.

However, if you have pushed the accelerator to the floor, that would be a positive acceleration, since it is accelerating your car in its direction of positive velocity.

The same concept is true of the train here. Since it is slowing down over some frame of time, and it is moving in the positive direction, the acceleration will be negative. Since we're talking about tangential components, the fact that the train is in rotational motion isn't even relevant.