**Problem Statement:**

The parametric equations (in m) of the trajectory of a particle are given by:

x(t) = 3t

y(t) = 4t^{2}

- Write the position vector of the particle in terms of the unit vectors.
- Calculate the velocity vector and its magnitude (speed).
- Express the trajectory of the particle in the form y(x)..
- Calculate the unit tangent vector at each point of the trajectory.
- Calculate the acceleration of the particle.

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**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 doesn’t 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.

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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 doesn’t depend on time; therefore the motion of this particle is **uniformly accelerated**.

Do no forget to include the units in the results.

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