To find the velocity of the car after 3 seconds, we need to integrate the acceleration function with respect to time to obtain the velocity function. Given the acceleration function a(t) = 2t^2 + 2, we can find the velocity function v(t) by integrating a(t):
v(t) = ∫[a(t)] dt
Integrating the acceleration function with respect to time, we get:
v(t) = ∫[(2t^2 + 2)] dt v(t) = (2/3)t^3 + 2t + C
Here, C represents the constant of integration. To determine the value of C, we use the initial velocity condition. Given that the initial velocity v(0) is 10 m/s, we can substitute t = 0 and v(t) = 10 into the velocity function:
10 = (2/3)(0)^3 + 2(0) + C 10 = C
Therefore, C = 10.
Now, we can substitute t = 3 into the velocity function to find the velocity after 3 seconds:
v(t) = (2/3)t^3 + 2t + 10 v(3) = (2/3)(3)^3 + 2(3) + 10 v(3) = (2/3)(27) + 6 + 10 v(3) = 18 + 6 + 10 v(3) = 34 m/s
Therefore, the car is going at a velocity of 34 m/s after 3 seconds.