Final answer:
The second order system's constants, in terms of the nonzero initial conditions x0 and v0, can be determined using the general solution tailored to those conditions and integrating when acceleration is constant to find velocity and position over time.
Step-by-step explanation:
Calculating Constants of a Second Order System
To calculate the constants (Magnitude and Phase) of a second order system with nonzero initial conditions (x0 and v0), we consider the general solution of the differential equation characterizing the system's motion. For a system with constant coefficients, the solution can be expressed in terms of sines and cosines, or exponentially for damped oscillations. The constants of integration in these solutions are then determined by the initial conditions. However, a non-homogeneous second order system includes an additional function that accounts for external forces or inputs, and its particular solution must be derived according to the form of the non-homogeneous term.
To derive the expression for a non-homogeneous system, we typically look for a particular solution that will satisfy the non-homogeneous portion of the differential equation and combine it with the homogeneous solution. The initial conditions are then applied to solve for the constants in the complete solution.
Solving for Final Position with Constant Acceleration
When examining motion under constant acceleration, the final position can be found by analyzing horizontal and vertical motion components and employing kinematic equations. If the acceleration is known to be constant, we can integrate the acceleration function to find velocity and position as functions of time (x(t) and v(t)).