Final answer:
To find the velocity and position functions from an acceleration function, the initial velocity and initial position are required. Integrating the acceleration function provides the velocity function, and integrating the velocity function provides the position function, both with the addition of their respective initial condition to solve the constants.
Step-by-step explanation:
When given the acceleration function to find the velocity function and position function, you need to know the initial velocity and initial position as additional information. The initial velocity is necessary because the velocity function at any given time can be found by integrating the acceleration function and adding the initial velocity. Without knowing the initial velocity, the integration process will result in an arbitrary constant, which can be resolved by having this initial condition. Similarly, integrating the velocity function gives us the position function up to an arbitrary constant, which is resolved using the initial position. Therefore, the correct answers are (a) Initial velocity and (d) Initial position.
To apply this in practice, use the integral formulation of the kinetic equations. When analyzing motion, the functional form of velocity versus time is found by integrating the acceleration function with respect to time, adding the initial velocity. Likewise, the functional form of position versus time is found by integrating the velocity function, adding the initial position. Examples include motion analysis where a(t) = pt² — qt³ and initial conditions are given as zero, then the velocity and position functions are found through integration.