A particle is moving with acceleration a(t) = 30 t + 12. its position at time t =0 is s(0) = 11 and its velocity at time t =0 is v(0) = 16. What is its position at time t = 8?
To find the position of the particle at time t = 8, we need to integrate the acceleration function to get the velocity function, and then integrate the velocity function to get the position function.
Given acceleration a(t) = 30t + 12, we can integrate it with respect to time to get the velocity function v(t):
∫a(t) dt = ∫(30t + 12) dt
Using the power rule of integration, we get:
v(t) = 15t^2 + 12t + C1
Where C1 is the constant of integration. To find C1, we can use the initial velocity v(0) = 16:
v(0) = 15(0)^2 + 12(0) + C1 = 16
C1 = 16
So, the velocity function becomes:
v(t) = 15t^2 + 12t + 16
Next, we integrate the velocity function to get the position function s(t):
∫v(t) dt = ∫(15t^2 + 12t + 16) dt
Using the power rule of integration, we get:
s(t) = 5t^3 + 6t^2 + 16t + C2
Where C2 is the constant of integration. To find C2, we can use the initial position s(0) = 11:
s(0) = 5(0)^3 + 6(0)^2 + 16(0) + C2 = 11
C2 = 11
So, the position function becomes:
s(t) = 5t^3 + 6t^2 + 16t + 11
Now, to find the position at time t = 8, we substitute t = 8 into the position function:
s(8) = 5(8)^3 + 6(8)^2 + 16(8) + 11
s(8) = 2560 + 384 + 128 + 11
s(8) = 3083
Therefore, the position of the particle at time t = 8 is 3083.