Final answer:
a. When the velocity is zero, the body's acceleration is 0 m/s^2. b. When the acceleration is zero, the body's speed is 9 m/s. c. The total distance traveled by the body from t = 0 to t = 2 is 10 m.
Step-by-step explanation:
a. To find the body's acceleration when the velocity is zero, we need to find the values of t when the velocity is zero. The velocity of the body is given by v(t) = s'(t), which is the derivative of the position function with respect to time. The acceleration of the body is given by a(t) = v'(t), which is the derivative of the velocity function with respect to time. So, first, we find the velocity function as v(t) = s'(t) = 3t^2 - 12t + 9
To find when the velocity is zero, we set v(t) = 0 and solve for t:
0 = 3t^2 - 12t + 9
By factoring: 0 = (t-1)(3t-9)
This gives us t = 1 and t = 3. So, when the velocity is zero, the body's acceleration is a(1) = 3(1)^2 - 12(1) + 9 = 0 m/s^2 and a(3) = 3(3)^2 - 12(3) + 9 = 0 m/s^2.
b. To find the body's speed when the acceleration is zero, we need to find the values of t when the acceleration is zero. The acceleration of the body is given by a(t) = v'(t), which is the derivative of the velocity function with respect to time. So, to find when the acceleration is zero, we need to find when v'(t) = 0. The acceleration function is a(t) = v'(t) = 6t - 12.
Setting a(t) = 0, we have 6t - 12 = 0. Solving for t gives t = 2. So, when the acceleration is zero, the body's speed is given by the magnitude of the velocity function at t = 2: |v(2)| = |3(2)^2 - 12(2) + 9| = 9 m/s.
c. To find the total distance traveled by the body from t = 0 to t = 2, we need to find the definite integral of the speed function from t = 0 to t = 2. The speed function is given by |v(t)| = |3t^2 - 12t + 9|.
Integrating the speed function over the interval [0,2]:
D = ∫[0,2] |v(t)| dt = ∫[0,2] (3t^2 - 12t + 9) dt = [(t^3 - 6t^2 + 9t)]_[0,2] = (2^3 - 6(2)^2 + 9(2)) - (0^3 - 6(0)^2 + 9(0)) = 10 m.