Question

If r⃗ =bt2i^+ct3j^, where b and c are positive constants, when does the velocity vector make an angle of 45.0∘ with the x- and y-axes?

Answer 1

The displacement function is given by : r = bt2i + ct3j meters
The velocity function is the derivative of the displacement:
v = r' = 2bti + 3ct2j meter / second

Now, for  the velocity vector make an angle of 45.0∘ with the x- and y-axes, the i and j components have to be equal:
2bt = 3ct2 (Now solve for t)
2bt = t (3ct)
either t = zero (rejected)
or t = 2b / 3c seconds (accepted)