I'm interested in equations of motion when friction is present for a little graphical side project I am working on. I'm really rusty with physics, so I apologize if this is a basic question or doesn't make sense...
Consider an object on a plane in three dimensions. It has initial position $ \vec{x_0} $ on the plane and initial velocity $ \vec{v_0} $ tangential to the plane. Assume it has acceleration (i.e. downhill gravity), $ \vec{a_G} $, which is constant and tangential to the plane. Now assume it has an additional acceleration (from friction, which I'm modeling as an acceleration), $\vec{a_F}(t) $, with constant magnitude $ k $ but variable direction, opposite that of velocity.
Is there a function $ \vec{x}(t) $ that gives position with respect to time? Something in the vein of the $$ x(t) = x_0 + v_0*t + 0.5*a*t^2 $$ that I learned in high school?