Find the position function s(t) given acceleration a(t)=3t if v(2)=0 and s(2)=1.
a(t) = 3t
v(t) = (3/2)t^2 + c
v(2) = 0 ----> 0 = (3/2)(4) + c
0 = 6 + c
c = -6
v(t) = (3/2)t^2 - 6
s(t) = (1/2)t^3 - 6t + k
s(2) = 1 -----> 1 = 4 - 12 + k
k = 9
s(t) = (1/2)t^3 - 6t + 9
To find the position function, we need to integrate the acceleration function twice.
Given the acceleration function a(t) = 3t, we can integrate it once to find the velocity function v(t) and then integrate it again to find the position function s(t).
Step 1: Integrate the acceleration function to find the velocity function:
∫a(t) dt = ∫3t dt
= (3/2) t^2 + C
where C is the constant of integration.
Step 2: Use the given information v(2) = 0 to solve for the constant of integration:
v(2) = (3/2) (2)^2 + C = 0
=> (3/2) (4) + C = 0
=> 6 + C = 0
=> C = -6
So, the velocity function v(t) = (3/2) t^2 - 6.
Step 3: Integrate the velocity function to find the position function:
∫v(t) dt = ∫[(3/2) t^2 - 6] dt
= (1/2) (3/2) t^3 - 6t + C
Step 4: Use the given information s(2) = 1 to solve for the constant of integration:
s(2) = (1/2) (3/2) (2)^3 - 6(2) + C = 1
=> (1/2) (3/2) (8) - 12 + C = 1
=> (3)(4/2) - 12 + C = 1
=> 6 - 12 + C = 1
=> -6 + C = 1
=> C = 7
So, the position function s(t) = (1/2) (3/2) t^3 - 6t + 7.
To find the position function s(t) given the acceleration function a(t) and the initial conditions, we need to integrate the acceleration function twice with respect to time.
First, let's integrate the acceleration function a(t) to find the velocity function v(t):
∫a(t) dt = ∫3t dt
v(t) = 3/2 * t^2 + C1
We are given that v(2) = 0, so we can use this information to find the constant C1:
v(2) = 3/2 * (2)^2 + C1
0 = 3/2 * 4 + C1
C1 = -6
Now we have the velocity function v(t):
v(t) = 3/2 * t^2 - 6
Next, we integrate the velocity function v(t) to find the position function s(t):
∫v(t) dt = ∫ (3/2 * t^2 - 6) dt
s(t) = 1/2 * t^3 - 6t + C2
We are given that s(2) = 1, so we can use this information to find the constant C2:
s(2) = 1/2 * (2)^3 - 6(2) + C2
1 = 1/2 * 8 - 12 + C2
1 = 4 - 12 + C2
C2 = 9
Finally, we have the position function s(t):
s(t) = 1/2 * t^3 - 6t + 9
So, the position function s(t) given the provided acceleration function and initial conditions is s(t) = 1/2 * t^3 - 6t + 9.