91Ó°ÊÓ

The process of finding velocity vectors from acceleration vectors involves integrating the components of the acceleration vector. Given an acceleration vector \( \mathbf{a}(t) = \mathbf{i} + e^{-t} \mathbf{j} \), you need to determine the velocity vector \( \mathbf{v}(t) \). This is achieved by integrating each component of the acceleration vector separately.- Integrating the \( \mathbf{i} \) component, which is constant (1), leads to \( v_x(t) = \int 1 \, dt = t + C_1 \), representing motion in the x-direction.- For the \( \mathbf{j} \) component, \( v_y(t) = \int e^{-t} \, dt = -e^{-t} + C_2 \), which describes motion in the y-direction.Once integrated, these components combine to form the velocity vector \( \mathbf{v}(t) = (t + C_1)\mathbf{i} + (-e^{-t} + C_2)\mathbf{j} \). This method of integration is fundamental to understanding how to translate acceleration into velocity, a crucial step in solving any problem involving motion dynamics.

Initial conditions provide the specific values that allow us to find unique solutions to differential equations. In this exercise, two initial conditions are given: \( \mathbf{v}(0) = 2\mathbf{i} + \mathbf{j} \) and \( \mathbf{r}(0) = \mathbf{i} - \mathbf{j} \).- For the velocity vector, substituting \( t = 0 \) gives \( \mathbf{v}(0) = C_1\mathbf{i} + (C_2 - 1)\mathbf{j} \). Using the initial condition \( 2\mathbf{i} + \mathbf{j} \), we can solve for the constants, yielding \( C_1 = 2 \) and \( C_2 = 2 \).- Similarly, for the position vector, substituting \( t = 0 \) in \( \mathbf{r}(t) \), we obtain \( C_3\mathbf{i} + (1 + C_4)\mathbf{j} \). Matching it with the given initial condition \( \mathbf{i} - \mathbf{j} \), results in \( C_3 = 1 \) and \( C_4 = -2 \).These initial conditions ensure that both the velocity and position vectors accurately reflect the motion of the particle from its starting configuration.

During integration, constants of integration (e.g., \( C_1, C_2, C_3, \) and \( C_4 \)) arise. These constants reflect unknown factors that can be determined using initial conditions. They essentially allow the integration process to translate general solutions into specific solutions that fit the unique situation described in a problem.- For the velocity, the integration of \( a_x(t) \) and \( a_y(t) \) results in constants \( C_1 \) and \( C_2 \), which are placeholders until initial conditions provide their values.- Similarly, the integration to find the position vector introduces other constants \( C_3 \) and \( C_4 \).Once initial conditions are applied, these constants are no longer arbitrary and serve to provide a complete, individualized solution for the motion of the particle. Understanding and correctly identifying these constants are vital for accurately applying calculus to problems in physics and engineering, particularly in mechanics, where motion analysis is key.