This question is from dynamics can you please explain in det

This question is from dynamics, can you please explain in detail. I can\'t find this in the book.

Which of the following can be negative?

Solution

The tangential unit vector points always in the same direction as the velocity: it is defined as the velocity vector divided by the magnitude of velocity. If the speed increases, the tangential acceleration is positive. In case of decreasing speed, the tangential acceleration is negative. In the equation |v| is the speed, and d|v|/dt can be both positive and negative.

The direction of tangential unit vector is the same as that of the velocity. So the velocity is always positive in the coordinate system defined by radial and tangential unit vectors, moving together with the body. The body can accelerate along the tangent in the direction of the velocity, then the magnitude of the velocity will increase; or it can accelerate in the opposite direction (the acceleration is negative, the body decelerates), the magnitude of velocity will decrease. Imagine a car travelling along a circular rod. It starts from rest and accelerates - its velocity and tangential acceleration are of the same direction, the tangential acceleration is positive with respect to the direction of travel. Suddenly a deer jumps out on the road, so the driver has to brake. Now the acceleration is opposite to the direction of travel, it is negative, as the car has to slow down.