# Motion

Motion is defined as the change in the position of an object over time and mathematically described in terms of displacement, distance, velocity, acceleration, speed, and time. When we talk about motion of points, we refer to points belonging to rigid bodies, intending to establish the connection between the motions of the two entities – point and body – as determined by the position and constraints. The motion of a body is observed by attaching a frame of reference to an observer and measuring the change in position of the body relative to that frame.

If the position of an object is not changing relatively to a given frame of reference, the object is said to be at rest, motionless, immobile, stationary, or to have a constant or time-invariant position with reference to its surroundings. As there is no absolute frame of reference, absolute motion cannot be determined. Thus, everything in the universe can be considered to be in motion.

## Types of motion of a body

There are various types of motion: rectilinear, curvilinear, circular, parabolic and elliptical: all these are called trajectories. By composing the individual motions together, it is possible to determine the trajectory of a body in space. Among the motions of classical physics, we remember:

• Translation
• Rotation
• Roto-translation
• Plane motion
• Uniform motion
• Uniform straight motion
• Smoothly accelerated motion
• Relative motion
• Cyclical motion
• Circular motion
• Transient motion
• Anharmonic motion

### Translation

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or space by the same distance in a given direction. It can also be interpreted as the addition of a constant vector at each point, or as a displacement of the origin of the coordinate system.

In physics, translation is defined as the act of motion of a material body subject to an action such as to cause a displacement on a straight path. The translation is a special case of rotation around an instant center of rotation infinitely distant, located in the direction perpendicular to that of translation. In other words, it is as if a body is moving on a circular trajectory with an infinite radius.

Known the motion of the center of mass, the positions of all points are also known because the distances are fixed. The translation is described by three degrees of freedom, which correspond to the coordinates of the center of mass. In other words, the translational motion of a rigid body has three degrees of freedom described by the three coordinates of the position vector of the center of mass.

### Rotation

Rotation is defined as rigid movement having as fixed points a point called center (in two dimensions) or a straight line called axis (in three dimensions) of rotation. This movement shifts all points around the center, or axis, by a fixed angle. In other words a rotation is the movement of a body following a circular trajectory. In two dimensions, ie on the plane, a figure can rotate around a point called the instant rotation center; in three dimensions, the rotation takes place around a straight line called the instant rotation axis and more generally, a rotation in $$n$$ dimensions takes place around a space with $$n-2$$ dimensions.

This formalizes the sensory perception according to which rotating a non-deformable object is equivalent to varying the angle of all its points by the same quantity, ie leaving the reciprocal angular relations unchanged. In particular, a rotation is a transformation that preserves the scalar product.

### Roto-translation

The rigid transformation (or roto-translation motion) is the composition between reflection, translation, and rotation, and therefore it is an isometry, that is, a geometric transformation that leaves the distances unchanged. In other words, we can think of roto-translation as a rigid movement in which a geometric figure first rotates and then translates.

The rototranslation motion of a rigid body has 6 degrees of freedom (3 of translation and 3 of rotation). To define a rototranslation we need:

• a point in the plane or a straight line in space with respect to which rotate;
• an angle $$\alpha$$ characterized by an amplitude and a direction (clockwise or counterclockwise);
• a vector $$\vec{v}$$ against which to translate.

It is important to respect the order in which the two transformations are carried out: first rotate and then translate. If the order is reversed and is executed before the translation and then the rotation, it could happen to get with a figure that has a different position than it should have.

### Relative motion

Relative motion is the study of the motion of an object with regard to some other moving object. Thus, the motion is not calculated with reference to the earth but is the velocity of the object in reference to the other moving object as if it were in a static state. Normally the reference system used is the Cartesian plane (or a three-dimensional Cartesian coordinate system) or a polar coordinate system (because depending on the case it may be more useful). The laws of physics which apply when you are at rest on the earth also apply when you are in any reference frame which is moving at a constant velocity with respect to the earth. The importance of relative motions concerns the correct application of the laws of kinematics and dynamics with respect to the reference system studied.

From the point of view of physics, to say that a body has a certain speed value or is located in a particular position is incomplete information if we do not specify compared to what we measured the speed or that position. All motions are relative motions with respect to a fixed reference system: that is, we must always specify the reference system used to describe the physical phenomena we are studying. Depending on the reference system chosen based on the needs and convenience of the model we want to study, the position and speed change: the motion is therefore always relative. Different observers in different reference systems describe what they see differently.

The velocity of the moving objects with respect to other moving or stationary object is called “relative velocity” and this motion is called “relative motion”. The motion may have a different appearance as viewed from a different reference frame, but this can be explained by including the relative velocity, relative speed, or relative acceleration (which is the change in velocity divided by the change in time) of the reference frame in the description of the motion.

Even the trajectory of a body changes in relative motions and according to the reference system from which its motion is observed. For example, when it snows and we look at the flakes as they fall from the window, we see them fall vertically; however, if we find ourselves traveling by car, the flakes go towards the windscreen with a certain inclination. If a person traveling in a train wagon throws a ball upwards and then picks it up again, for him the ball will have made a simple vertical motion in free fall, but for an observer who is stationary on the platform, the ball will have a parabolic movement because the ball, in addition to moving vertically, also moved horizontally with the same speed as the train.

#### Relative motion in one dimension

In one-dimensional motion (in the classical or non-relativistic, or the Newtonian approximation) objects move in a straight line at speeds much less than the speed of light. So there are only two possible cases:

1. objects are moving in the same direction;
2. objects are moving in the opposite direction.

#### Relative motion in two dimensions

The same concept will be applicable in two-dimensional motion. If you have to find the velocity of A with respect to B, assume that B is at rest and give the velocity of B to A in the opposite direction. Let us consider two objects A and B which are moving with velocities $$V_a$$ and $$V_b$$ with respect to some common frame of reference, say, with respect to the ground or the earth. We have to find the velocity of A with respect to B, so assume that B is at rest and give the velocity of B to A in the opposite direction.

$V_{ab} = v_a – v_b$

Similarly, for the velocity of object B with respect to A, assume that A is at rest and give the velocity of A to B in the opposite direction.

$V_{ba} = v_b – v_a$