Question

A bullet moves in a circular path and its rotational velocity w = πt + 3 / 2π after t = 0, how many seconds does it take for the bullet to complete one round?

Answer 1

With given rotational velocity the covered angle is:

$$\varphi(t) = \int_0^t\omega(t)dt = \int_0^t\pi t+1.5\pi dt =\dfrac{\pi t^2}{2} + 1.5\pi t$$

Here we assumed that $\varphi(0) = 0$. After one revolution, the agle becomes $\varphi(t) = 2\pi$ substituting it into the formula above obtain the equation for $t$:

$$\dfrac{\pi t^2}{2} + 1.5\pi t = 2\pi\\ 0.5t^2+1.5t-2 = 0\\ t^2 +3t-4 = 0$$

The roots are:

$$t_1 = 1, t_2 = -4$$

Taking the positive root (since we consider time after t=0), find the final answer.

Answer. 1s.