To find the initial acceleration, we need to differentiate the position function with respect to time and evaluate it at the initial time (t = 0).
Given the position function:
s = 3t + ⅔t² - 2t³
To find the velocity function, differentiate the position function with respect to time:
v = ds/dt
Taking the derivative of each term separately:
v = d/dt(3t) + d/dt(⅔t²) - d/dt(2t³) v = 3 + ⅔(2t) - 3(2t²)
Simplifying:
v = 3 + ⅔(2t) - 6t²
Now, to find the acceleration function, differentiate the velocity function with respect to time:
a = dv/dt
Taking the derivative of each term separately:
a = d/dt(3 + ⅔(2t) - 6t²) a = 0 + ⅔(2) - 6(2t)
Simplifying:
a = ⅔ - 12t
To find the initial acceleration, substitute the initial time (t = 0) into the acceleration function:
a_initial = ⅔ - 12(0) a_initial = ⅔
Therefore, the initial acceleration is ⅔.