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A vector A varies with time as A(t) = t * i + 2*t² * j (where i and j are unit vectors along x and y axes). Find the rate of change of the angle that A makes with the positive x-axis, at t = 1/2 s. (Answer in rad/s.)
- 1 rad/s
- 2 rad/s
- 1/2 rad/s
- 4 rad/s
Correct answer: 1 rad/s
Solution
The angle the vector makes with the x-axis is theta(t) = arctan(Ay/Ax) = arctan(2t²/t) = arctan(2t). Differentiating: d(theta)/dt = 2/(1+(2t)²). At t = 1/2: d(theta)/dt = 2/(1 + 4*(1/4)) = 2/(1+1) = 2/2 = 1 rad/s.
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