A particle moves according to a law of motion s = f(t), t ≥ 0, where t is measured in seconds and s in feet. f(t) = t3 − 12t2 + 36t (a) Find the velocity at time t. v(t) = 3t2−24t+36 (b) What is the velocity after 5 s? v(5) = ft/s (c) When is the particle at rest? (Enter your answer as a comma-separated list.) t = 6,2 (d) When is the particle moving in the positive direction? (Enter your answer in interval notation.) t = (2,6) (e) Find the total distance traveled during the first 8 s. ft

a) is the velocity of the particle at any time t.

b)

c) The particle is at rest at time t=2 seconds.

d) At time t=2 seconds the particle posses positive velocity being at positive direction.

e)

Explanation:

Given:

The function of displacement dependent on time, .......(1)

a)

Now as we know that velocity is the time derivative of the displacement:

..........................(2)

b)

Now the velocity after 5 seconds:

put t=5 in eq. (2)

c)

When the particle is at rest it has zero velocity.

Now velocity at time t=6 s:

Now velocity at time t=2 s:

The particle is at rest at time t=2 seconds.

d)

Put the value t=2 sec. in eq. (1):

At time t=2 seconds the particle posses positive velocity being at positive direction.

e)

Distance travelled during the first 8 seconds:

put t=8 in eq. (1)

v=ir = 3x15 =45 v