Confused about Centripetal and Radial Acceleration

[JustSomeGuy80]

Hello, I was trying to do this problem and then I looked at the solution manual and found something that confused me. I am having trouble distinguishing between centripetal and radial acceleration. According to the equation in the book $$a_r=-a_c$$, which kind of confuses me. Isn't a centripetal acceleration vector pointed towards the center of a circle in uniform circular motion? Then why do they draw the radial acceleration vector, which has the opposite sign of the centripetal acceleration vector, pointing towards the center of the circle also? Aren't they supposed to be pointing in opposite directions? In the solution manual, he draws the radial acceleration vector, but then uses the centripetal equation for the solution. Can someone explain this to me? Particularly how centripetal and radial acceleration relate to one another. An analogy would nice if possible. Thx.

Here is a http://i218.photobucket.com/albums/cc304/JustSomeGuy805/PHYSCS.jpg" of what I am talking about where the radial acceleration vector is drawn but then the centripetal acceleration equation is used instead.

Homework Statement

A train slows down as it rounds a sharp horizontal turn, slowing from 90.0 km/h to 50.0 km/h in the 15.0 seconds that it takes to round the bend. The radius of the curve is 150 m. Compute the acceleration at the moment the train speed reaches 50.0 km/h. Assume that it continues to slow down at this time at a constant rate.

Homework Equations

v=velocity, a=acceleration, r=radius
radial acceleration: $${a_r} = - {{{v^2}} \over r}$$


tangential acceleration: $${a_t} = \left| {{{dv} \over {dt}}} \right|$$

acceleration vector: $$\vec a = {{\vec a}_r} + {{\vec a}_t}$$


centripetal acceleration: $${a_c} = {{{v^2}} \over r}$$

 

[PhanthomJay]

The total acceleration is the vector sum of the tangential acceleration and the acceleration perpendicular to the tangential acceleration, which is the radial accelearation. The radial acceleration and the centripetal acceleration are one and the same; both point inwards toward the center of the circle. I don't know why they put a minus sign on the centripetal acceleration, unless they got confused with centrifugal acceleration, which does not exist in an inertial reference frame.