I usually find it easiest to put the dependency on t into the object or not as in
In[1]:= a Cos[alpha[t]]
Out[1]= a Cos[alpha[t]]
In[2]:= D[%, t]
Out[2]= -(a Sin[alpha[t]] alpha'[t])
But you can also just define the derivative of a to be zero with respect to t before you do anything.
