in some cases we use dv/dr to find speed and in some cases we use v/r which 1 will be more appropriate to use?
Er.. you can't use either one!
Speed, or velocity, is defined as ds/dt, where s is displacement. If speed is constant, then finding the total net displacement S, over a period of time T, will give you v = S/T (for v constant). If v isn't constant, then the instantaneous you use ds/dt at a particular time t.
What are v and r? If they are distance and time , then dv/dr is the appropriate quantity. v/r will work for constant velocities only.
However if v and r represent velocity and displacement and you are talking about angular velocity then v/r is the correct quantity.
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