A particle starts from rest and moves in a straight line such that its acceleration after t secs is given by \(a = (3t - 2) ms^{-2}\). Find the distance cov...

To solve this problem, we need to use the kinematic equation: $$s = ut + \frac{1}{2}at^2$$ where: - s is the distance covered - u is the initial velocity (which is zero in this case) - a is the acceleration - t is the time taken We are given that the acceleration is given by \(a = (3t-2)ms^{-2}\). Integrating this with respect to time, we get: $$v = \int a dt = \int (3t-2) dt = \frac{3}{2}t^2 - 2t + C$$ where v is the velocity of the particle at time t, and C is a constant of integration. Since the particle starts from rest, we have v = 0 when t = 0, so: $$0 = \frac{3}{2}(0)^2 - 2(0) + C$$ which gives C = 0. Therefore, the velocity of the particle at time t is given by: $$v = \frac{3}{2}t^2 - 2t$$ Now, we can use the equation for distance covered to find the distance covered by the particle after 3 seconds: $$s = ut + \frac{1}{2}at^2 = 0 + \frac{1}{2}(3t-2)t^2 = \frac{3}{2}t^3 - t^2$$ So, when t = 3, we have: $$s = \frac{3}{2}(3)^3 - (3)^2 = \frac{27}{2} - 9 = \frac{9}{2}m$$ Therefore, the distance covered by the particle after 3 seconds is \(\frac{9}{2}m\). Thus, the correct option is: - \(\frac{9}{2}m\)