# The position of a particle moving along an x axis is given by x = 16.0t2 - 6.00t3, where x is in meters and t is in seconds.?

x = 16 * t^2 – 6 * t^3

x = 16 * 3^2 – 6 * 3^3 = -18 meters

To determine the equation of velocity versus time, take the first derivative of the equation above.

vf = vi + 32 * t – 18 * t^2, vi = 0

vf = 32 * t – 18 * t^2

vf = 32 * 3 – 18 * 3^3 = -66 m/s

The equation of acceleration versus time is the first derivative of this equation.

a = 32 – 36 * t

a = 32 – 36 * 3 = -7.6 m/s

The maximum positive velocity is when the acceleration is 0.

a = 32 – 36 * t = 0

t = 8/9 second

v = 32 * 8/9 – 18 * (8/9^2

This is 14 2/9th m/s.

(h) What is the acceleration of the particle at the instant the particle is not moving (other than at t = 0)?

Let’s use the following equation to determine the when the velocity is 0 m/s

vf = 32 * t – 18 * t^2

0 = 32 * t – 18 * t^2

18 * t^2 = 32 * t

t = 32/18

This is 1 7/9th second.

a = 32 – 36 * 32/18 = -32 m/s

i) Determine the average velocity of the particle between t = 0 and t = 3.00 s.

At 3seconds, the velocity is -66 m/s.

v ave = ½ * (vi + vf), vi = 0

v ave = -33 m/s

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