Position Velocity Acceleration vectors - Parametric equations

Solution:

The position vector is given by:

Where i y j are the unit vectors defining the positive direction of the x and y axes respectively. Since the position vector does not have z component, the particle describes a two dimensional trajectory.

The velocity vector is given by:

And its magnitude (speed):

To express the trajectory of the particle in the form y(x) we first isolate the variable (t) in the first equation x(t) and substitute into the second:

You can find below a plot of the trajectory, which is a parabola.

In the plot below you can see the tree vectors (r, v y a) which allow to describe the motion of a particle.

To calculate the tangent unit vector u_t we have to transform the velocity vector into a parallel vector whose magnitude is one (remember that the velocity vector is always tangent to the path of motion). This is done by dividing the velocity vector by its magnitude:

This tangent unit vector depends on time because, even if its magnitude is always unity, its direction varies for a curvilinear path.

The acceleration vector is defined:

Therefore, deriving the velocity vector:

The acceleration vector does not depend on time; therefore the motion of this particle is uniformly accelerated.

Do no forget to include the units in the results.