Suppose the position of an object is given by r=(3.0t^2hat (i)-6t^3j) m. (a) Determine its velocity v and acceleration a as a function of time. (b) Determine r and v at time t=2.5s

(a) To determine the velocity v and acceleration a of the object, we need to take the derivatives of the position function r with respect to time t.<br /><br />The position function is given by:<br />r = (3.0t^2 hat{i} - 6t^3 hat{j}) m<br /><br />To find the velocity, we take the derivative of r with respect to t:<br />v = dr/dt = (6.0t hat{i} - 18t^2 hat{j}) m/s<br /><br />To find the acceleration, we take the derivative of v with respect to t:<br />a = dv/dt = (6 hat{i} - 36t hat{j}) m/s^2<br /><br />Therefore, the velocity v and acceleration a as a function of time are:<br />v = (6.0t hat{i} - 18t^2 hat{j}) m/s<br />a = (6 hat{i} - 36t hat{j}) m/s^2<br /><br />(b) To determine r and v at time t = 2.5 s, we substitute t = 2.5 s into the expressions for r and v.<br /><br />For r:<br />r = (3.0(2.5)^2 hat{i} - 6(2.5)^3 hat{j}) m<br />r = (3.0(6.25) hat{i} - 6(15.625) hat{j}) m<br />r = (18.75 hat{i} - 93.75 hat{j}) m<br /><br />For v:<br />v = (6.0(2.5) hat{i} - 18(2.5)^2 hat{j}) m/s<br />v = (15 hat{i} - 18(6.25) hat{j}) m/s<br />v = (15 hat{i} - 112.5 hat{j}) m/s<br /><br />Therefore, at time t = 2.5 s, the position r and velocity v are:<br />r = (18.75 hat{i} - 93.75 hat{j}) m<br />v = (15 hat{i} - 112.5 hat{j}) m/s