Step-by-step explanation:
To find the position function x(t), we need to integrate the acceleration function twice with respect to time.
Since the acceleration is constant, we can use the kinematic equation:
v(t) = v0 + a*t
where v0 is the initial velocity.
At t = 0.13 s, we have:
v(0.13) = -0.185 m/s
a = 0.899 m/s^2
So,
v(t) = -0.185 + 0.899t
Integrating v(t) with respect to time gives:
x(t) = x0 + ∫v(t) dt
where x0 is the initial position. Since we don't know the initial position, we can set x0 = 0 without loss of generality.
Integrating v(t) gives:
x(t) = ∫(-0.185 + 0.899t) dt = -0.185t + 0.4495t^2 + C1
where C1 is the constant of integration.
To find C1, we use the initial condition x(0.13) = 0:
0 = -0.185(0.13) + 0.4495(0.13)^2 + C1
C1 = 0.00922
So, the position function is:
x(t) = -0.185t + 0.4495t^2 + 0.00922