Final answer:
To determine the function s(t), integrate the velocity function v(t) and use initial conditions. For the given v(t) and a(t), set v(t) to zero and solve for t to find when the object is momentarily at rest, resulting in t values of 0 or approximately 0.83 seconds.
Step-by-step explanation:
To find the function s(t) that satisfies the given conditions, we need to integrate the velocity function v(t). First, we must identify the known quantities, such as the initial conditions for position and velocity. This helps us to determine the appropriate physical relationships and equations to use. In the case provided, we have a velocity function given by v(t) = 10t - 12t² m/s, and an acceleration function a(t) = 10 - 24t m/s².
By integrating the velocity function, we find the position function s(t), using the initial conditions to solve for the constants of integration. To find the time t when certain conditions are met, such as the velocity being zero, we set v(t) = 0 and solve for t. For example, with the given functions, setting v(t) = 0 results in two possible solutions for t: 0 seconds or approximately 0.83 seconds. The latter gives a position x = 1.16 m and is the correct choice as per the given problem context.