Question

Determine the position function s(t) at t = 1 of a moving object if the acceleration is given by a(t) = 12t - 4 and v(2) = 4. s(0) = 1 -10 -11 10 11.

Answer 1

To determine the position function s(t) at t = 1 for the given acceleration, integrate the acceleration function to obtain the velocity function, and apply the initial condition to find the constant of integration. Then, integrate the velocity function to find the position function and apply initial conditions to solve for another constant. Finally, evaluate s(t) at t = 1.

Step-by-step explanation:

To find the position function s(t) at t = 1 for a moving object, given the acceleration function a(t) = 12t - 4 and initial conditions v(2) = 4 and s(0) = 1, we follow these steps:

1. Integrate the acceleration function a(t) to find the velocity function v(t). The indefinite integral of a(t) is:
2. ∑ a(t) dt = ∑ (12t - 4) dt = 6t^2 - 4t + C
3. Using the initial condition v(2) = 4, solve for the constant C.
4. 6(2)^2 - 4(2) + C = 4 → C = -16
5. The velocity function is therefore:
6. v(t) = 6t^2 - 4t - 16
7. Now integrate the velocity function to find the position function s(t).
8. ∑ v(t) dt = ∑ (6t^2 - 4t - 16) dt = 2t^3 - 2t^2 - 16t + D
9. Using the initial condition s(0) = 1, solve for the constant D.
10. 2(0)^3 - 2(0)^2 - 16(0) + D = 1 → D = 1
11. The position function is:
12. s(t) = 2t^3 - 2t^2 - 16t + 1
13. Finally, evaluate the position function at t = 1:
14. s(1) = 2(1)^3 - 2(1)^2 - 16(1) + 1 = 2 - 2 - 16 + 1 = -15