Vector Valued Functions and Motion in Space
The position of a particle in the xy plane at time t is Find an equation in x and y whose graph is the path of the particle. Then find the particle's velocity and acceleration vectors at t = 1.
To solve this problem, the first step is to find an equation that describes the path of the particle in the xy plane. The position vector is given in terms of time, with x and y expressed as functions of t. The goal here is to eliminate the parameter t and find a relationship between x and y directly, which will give you the particle's trajectory. This usually involves solving one of the parametric equations for t and substituting it into the other equation.
Next, to find the velocity vector, recall that velocity is the rate of change of the position with respect to time. This means you need to differentiate the position vector with respect to t. The components of the velocity vector will come from taking the derivative of the x and y functions.
Finally, to find the acceleration vector, differentiate the velocity vector with respect to time. This gives the rate at which the velocity changes, or the acceleration. To find the specific values for the velocity and acceleration at t = 1, simply substitute t = 1 into the expressions you found for both the velocity and acceleration vectors.