Final answer:
To find the position of a particle given its acceleration function a(t) = t² - 5t + 4, and initial conditions s(0) = 0, s(1) = 20, we need to integrate the acceleration function twice to get the velocity and then the position, applying the initial conditions to solve for constants of integration.
Step-by-step explanation:
The question asks us to find the position of a particle whose acceleration function (a(t)) is given as t² - 5t + 4, and also provides initial conditions for the position function s(t) at two different times, s(0) = 0 and s(1) = 20. To find the position function, we need to integrate the acceleration function twice. The first integral will provide us the velocity function (v(t)), and the second integral will give us the position function (s(t)).
First, we integrate the acceleration function to find the velocity:
- ∫ a(t) dt = ∫ (t² - 5t + 4) dt = (1/3)t³ - (5/2)t² + 4t + C
Velocity function will have a constant of integration C, which we can determine using the initial velocity condition if it was provided. However, since it isn't provided, we'll move on.
Then, we integrate the velocity function to find the position:
- ∫ v(t) dt = ∫ ((1/3)t³ - (5/2)t² + 4t) dt = (1/12)t⁴ - (5/6)t³ + 2t² + Ct + D
The position function has another constant of integration D. The initial conditions s(0) = 0 and s(1) = 20 will help us find the constants C and D.