The position of a particle moving along an x axis is given by x = 14.0t2 - 4.00t3, where x is in meters and t is in seconds. Determine (a) the position, (b) the velocity, and (c) the acceleration of the particle at t = 5.00 s. (d) What is the maximum positive coordinate reached by the particle and (e) at what time is it reached? (f) What is the maximum positive velocity reached by the particle and (g) at what time is it reached? (h) What is the acceleration of the particle at the instant the particle is not moving (other than at t = 0)? (i) Determine the average velocity of the particle between t = 0 and t = 5.00 s.
Explanation / Answer
Given position of the partical moving in x- direction x = 14 t^2 - 4 t^3 velocity of the partical v = d x /dt = 28 t - 12 t^2 acceleration of the partical a = dv /dt = 28 - 24t position , velocity and acceleration of the particle at t = 5 s a) x = 14 ( 25 ) - 4 ( 125 ) = 350- 500 = - 150 m b) velocity v = 28 ( 5 ) - 12 ( 25 ) = - 160 m/s c) acceleration a = 28 - 24 ( 5) = -92 m/s ^2 maximum positive coordinate reached by the particle is at dx /dt = 0 28 t - 12 t^2 = 0 28 - 12 t = 0 t = 28 / 12 = 2.33 s x = 14 t^2 - 4 t^3 = 14*(2.33)^2- 4*(2.33)^3 = 25.4 m maximum positive velocity reached by the particle is at dv /dt = 0 28 - 24 t = 0 t = 28 / 24 = 1.166 v = 28 t - 12 t^2 = 28 * 1.166 - 12 ( 1.166)^2 = 16.333 m/s at t = 0 v = 0 average velocity v = - 150 m / 5 = - 30 m/s