Why Can Tangential Acceleration Vector's Scalar Component Not Be Negative?

[mike 2012]

we already know that the scalar component of the Centripetal ( Radial ) Acceleration vector is always negative because it's ALWAYS directed to the opposite direction of its unit vector ( toward the center of the circle ) , and this is satisfying to me and to the formula .

however , when it comes to the scalar component of the Tangential Acceleration vector , I get confused because its formula always gives positive outputs . which means it's always directed to the same direction of its unit vector .

the question is : why can the scalar component of the Tangential Acceleration vector not be negative and directed to the opposite direction of its unit vector ?

if it can be negative and directed to the opposite direction of its unit vector , why does not the formula satisfy negative outputs ?

http://imageshack.us/scaled/landing/26/hhhhjz.png

 

[ehild]

Hi Mike, welcome to PF.

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 you quoted |v| is the speed, and d|v|/dt can be both positive and negative.

ehild

 

[mike 2012]

The tangential unit vector points always in the same direction as the velocity

is this correct only for " the tangential unit vector " ?
because according to what I learned , any vector can be in the opposite direction of its unit vector and specified by a negative sign.

however , in case of " the tangential unit vector " , as I understood from you ; the unit vector follows the direction of the vector ( wether it's acceleration or velocity as you mentioned ) which means whatever the tangential vector is , it's positive (since its unit vector follows its direction )

and thank you for responding :)

 

[ehild]

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 traveling 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.

ehild

 

[Nickie]

I would like to clarify that the scalar component of the tangential acceleration vector can indeed be negative. This is because the direction of the tangential acceleration vector is determined by the direction of the velocity vector, which can change in both positive and negative directions depending on the motion of the object.

The formula for tangential acceleration, a = rα, where r is the radius of the circle and α is the angular acceleration, gives positive outputs because it is based on the magnitude of the tangential acceleration, which is always positive. However, the direction of the vector itself can be either positive or negative, depending on the direction of the velocity and the direction of the angular acceleration.

In the case of circular motion, the tangential acceleration is always perpendicular to the radial acceleration, which is directed towards the center of the circle. This is why the scalar component of the radial acceleration vector is always negative, as it is directed in the opposite direction of its unit vector.

In summary, the scalar component of the tangential acceleration vector can be negative, but the magnitude of the vector itself is always positive. The formula for tangential acceleration gives positive outputs because it is based on the magnitude of the vector, not its direction. It is important to consider both the magnitude and direction of a vector when analyzing motion.