Answer:The acceleration when the velocity is 0 is 12 m/s^2.
Step-by-step explanation:
(a) To express the velocity and acceleration as functions of time (t) using the given equation of motion, we need to differentiate the equation with respect to time.
Given: s = t^3 - 12t
First, let's determine the velocity (v):
The derivative of s with respect to t gives us the velocity:
v(t) = ds/dt
Differentiating the equation of motion:
v(t) = d/dt(t^3 - 12t)
= 3t^2 - 12
Therefore, the velocity as a function of time is v(t) = 3t^2 - 12.
Next, let's calculate the acceleration (a):
The derivative of v with respect to t gives us the acceleration:
a(t) = dv/dt
Differentiating the velocity equation:
a(t) = d/dt(3t^2 - 12)
= 6t
Therefore, the acceleration as a function of time is a(t) = 6t.
(b) To determine the acceleration after 5 seconds, substitute t = 5 into the acceleration equation:
a(5) = 6(5)
= 30 m/s^2
Hence, the acceleration after 5 seconds is 30 m/s^2.
(c) To find the acceleration when the velocity is 0, set v(t) = 0 and solve for t:
3t^2 - 12 = 0
Solving this quadratic equation yields:
3t^2 = 12
t^2 = 4
t = ±2
Since we are considering t ≥ 0, we discard the negative value and select t = 2.
Substituting t = 2 into the acceleration equation:
a(2) = 6(2)
= 12 m/s^2
Therefore, the acceleration when the velocity is 0 is 12 m/s^2.