To determine when the motion of the particle changes direction, we need to find the points where the velocity of the particle changes sign. The velocity is the derivative of the position function with respect to time.
Given the equation of motion: s = (1/3)t^3 - 2t^2 + 3t
To find the velocity function, we differentiate the position function with respect to time: v = ds/dt = d/dt[(1/3)t^3 - 2t^2 + 3t]
Differentiating each term separately, we get: v = (d/dt)(1/3)t^3 - (d/dt)(2t^2) + (d/dt)(3t) v = (1/3)(3t^2) - 2(2t) + 3 v = t^2 - 4t + 3
To find the points where the velocity changes sign, we solve the equation: t^2 - 4t + 3 = 0
Factoring the equation, we have: (t - 1)(t - 3) = 0
Setting each factor equal to zero, we find: t - 1 = 0 or t - 3 = 0
Solving for t in each case, we get: t = 1 or t = 3
Therefore, the motion of the particle changes direction at t = 1 and t = 3.