Position, velocity and acceleration vectors

Kinematics is the branch of Mechanics that describes the motion of particles, objects or groups of objects. In these pages we will analyse the motion of a point particle in any trajectory like for instance the one shown in black in the figure below. For the sake of simplicity, a flat trajectory has been represented in the figure, but the trajectory can be three-dimensional.

A reference frame at rest and an observer O located at the origin of a Cartesian coordinate system are also shown in the previous figure. This reference frame is called inertial. The orientation of the three Cartesian axes is indicated by the unit vectors i, j and k respectively. We will describe the motion of the particle with respect to this reference frame.

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The motion of a particle is described by three vectors: position, velocity and acceleration.

The position vector (represented in green in the figure) goes from the origin of the reference frame to the position of the particle. The Cartesian components of this vector are given by:

The components of the position vector are time dependent since the particle is in motion. In order to simplify the notation we will often omit this dependence in the expressions of the vectors.

The velocity vector is the time derivative of the position vector:

Which can also be expressed as:

The velocity vector is always tangent to the trajectory of the particle at each point.

The acceleration vector is the time derivative of the velocity vector:

Which can also be expressed as:

The acceleration vector is the variation of the velocity vector over time. Therefore, it must always be directed towards the inside of the particle trajectory, as shown in the figure.

In these pages you will find numerous problems where you will learn to calculate these three vectors in different situations.


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The acceleration vector can be expressed as a function of its projections on a reference frame moving with the particle and with axes that are respectively tangent and perpendicular (or normal) for each point of its trajectory. These projections are called tangential acceleration and normal acceleration (or centripetal).

The previous figure represents the acceleration vector expressed as the sum of these components.

The tangential acceleration is given by:

Where ut is a unit vector tangent to the path at each point that is determined by dividing the velocity vector by its magnitude:

The tangential acceleration provides information about the variation of the velocity vector magnitude.

On the other hand, the normal acceleration (or centripetal) is given by:

Where un is a unit vector perpendicular to the path at each point and ρ is the radius of curvature of the path.

The normal acceleration provides information about the variation of the velocity vector’s direction. If a particle describes a straight line, the radius of curvature is infinite and therefore its normal acceleration is zero.

In the particular case of a circular path, the normal acceleration magnitude is:

The different problems you will find in these pages will help you to learn how to calculate the components of the acceleration vector.

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