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EXAMPLE 9 The position of a particle is given by the equation s = f(t) = t - 12+ + 36t where t is measured in seconds and s in meters. (a) Find the velocity at time t. (b) What is the velocity after 3 s? After 4 s? (c) When is the particle at rest? t =6 S = 0 (d) When is the particle moving forward (that is, in the positive direction)? (e) Draw a diagram to represent the motion of the particle. (f) Find the total distance traveled by the particle during the first eight seconds. (g) Find the acceleration at time t and after 4 s. (h) Graph the position, velocity, and acceleration functions for 0 s t s 5. = = 2 SOLUTION S =0 s = 32 -24 (a) The velocity function is the derivative of the position function. Video Example '() s = f(t) = +3 - 12t2 + 36t v (t) = as = 3t2 - 24t + 36 (b) The velocity after 3 s means the instantaneous velocity when t = 3, that is v(3) - ds dt It = 3 = 3(3)2 - 24(3) + 36 = m/s. The velocity after 4 s is V(4) = 3(4) 2 - 24(4) + 36 = m/s. (c) The particle is at rest when v(t) = , that is, 3t2 - 24t + 36 = 3(t2 - 8t + 12) = 3(t - 2)(t - 6) = and this is true when t = (smaller t-value) or t = (larger t-value). Thus the particle is at rest after [ ]s (smaller t-value) and after ( s (larger t-value). (d) The particle moves in the positive direction when v(t) [? Vo, that is, 3t2 - 24t + 36 = 3(t - 2)(t - 6) [? Vo. This inequality is true when both factors are positive (t> ) or when both factors are negative (t . It moves backward (in the negative direction) when