The position of a particle moving along an x axis is given by x = 12.0 t 2 - 2.0

Question

The position of a particle moving along an x axis is given by x = 12.0t2 - 2.00t3, 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

At t = 5s, by substituting we get:

a) x = 12 x 25 - 2 x 125 = 50m

b) v = dx/dt = 24t - 6t^2. So, substituting t = 5s we get: v = (-30) m/s

c) a = dv/dt = 24 - 12t. So substituting t = 5s we get: a = -36m/s^2

e) x is maximum when dx/dt = 0 => 24t - 6t^2 = 0 => t = 4s

d) Maximum coordinate = 12 x 4^2 - 2 x 4^3 = 64m

g) Maximum velocity is reached when dv/dt = 0 => 24 -12t = 0 => t = 2s

f) Maximum velocity is 24 x 2 - 6 x 4 = 24 m/s

h) Time when body is not moving => v = 0 => t = 4s. So, acceleration = 24 - 12 * 4 = -24m/s^2

i) Average velocity = distance covered/time taken = 50 / 5 = 10m/s

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