9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions

An initial value problem (IVP) is a mathematical problem that involves finding a solution for a differential equation that satisfies a given set of initial conditions. In physics and engineering, initial value problems are often used to model the motion of a particle in the plane.

Given a differential equation that describes the position of a particle moving in the plane and an initial condition that specifies the position of the particle at a certain point in time, an IVP allows us to determine an expression for the position of the particle as a function of time. This can be done by solving the differential equation using techniques such as separation of variables, integrating factors, and substitution.

Once we have an expression for the position of the particle, we can use derivatives to determine various other properties of the motion, such as velocity, speed, and acceleration. The velocity of a particle moving along a curve in the plane is given by the derivative of the position function with respect to time, and the speed of a particle is given by the magnitude of the velocity vector. The acceleration of a particle moving along a curve in the plane is given by the derivative of the velocity function with respect to time.

In the case of a curve defined using parametric or vector-valued functions, we can use the same process to determine the velocity, speed and acceleration. First, we must convert the parametric or vector-valued function to a Cartesian equation, then we can find the velocity by taking the derivative of the position function with respect to time, the speed by the magnitude of the velocity vector, and the acceleration by the derivative of the velocity function with respect to time. 

Remember from previous units that if we take the integral of the speed (the absolute value of velocity), we can find the distance traveled (imagine adding up all of the tiny instantaneous distances to find a total distance). 

This same concept applies in parametric equations, but since velocity is expressed as a vector, we need to take the integral of the magnitude of velocity. (In vector-valued functions, the magnitude is equivalent to the distance formula, which is essentially taking the absolute value of the vector.)