The position of a particle moving along an x axis is given by x = 14.0 t 2 - 4.0

Question

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 = 3.00 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 betweent = 0 and t = 3.00 s.

Explanation / Answer

Here ,

x = 14 t^2 - 4 t^3

at time t= 3 s

a) position , x = 14 t^2 - 4 t^3

x = 14 * 3^2 - 4 * 3^3

x = 18 m

the position of particle is 18 m

b) for the velocity

v = dx/dt

v = d/dt( 14 t^2 - 4 t^3)

v = 28 * t - 12 * t^2

at t = 3 s

v = 28 * 3 - 12 * 3^2

v = -24 m/s

the speed of particle is -24 m/s

c)

acceleration , a = dv/dt

a = d/dt(28 * t - 12 * t^2)

a = 28 - 24 * t

t = 3 s

a = 28 - 24 * 3

a = -44 m/s^2

the acceleration of the particle is -44 m/s^2

d) for the maximum positive corrdinate

v = 0

0 = 28 * t - 12 * t^2

t = 2.33 s

hence , postion = 14 * 2.33^2 - 4 * 2.33^3

postion = 24.4 m

maximum positive coordinate reached by the particle is 24.4 m

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