 # Rectilinear and Curvilinear Motion: the basic concept.

Rectilinear and curvilinear motion are two types of motion systems used in physics. Rectilinear motion is defined as the motion of an object that travels in a straight line. It is also referred to as linear motion. Curvilinear motion is a type of motion in which an object moves along a curved path rather than a straight line as observed in rectilinear motion.

## Rectilinear and curvilinear motion;

### Rectilinear motion: definition.

Rectilinear and curvilinear motion are two types of motion systems used in physics. Rectilinear motion is defined as the motion of an object that travels in a straight line. It is also referred to as linear motion. This type of motion can happen in one or more dimensions including horizontal, vertical or diagonal.

The motion of an object can be described using a single coordinate’s axis such as the x-axis or y-axis because the motion of the object is in one direction. Let us take some examples to describe the rectilinear motion.

### Examples: Image of a rocket launched vertically, an example of rectilinear motion/credit lumenlearning.com
• Free falling of an object:  It is the most common example of the said motion. When an object is dropped from a certain height, it falls vertically downward in a straight line towards the ground by the influence of the gravity of the earth. The object’s alignment remains on the y-axis and can be described as taking a single coordinate axis.
• A pendulum swinging back and forth in a straight line.
• A boy running on a straight road.
• A rocket launched vertically upward into space.
• A car is moving on a straight highway.
• A train running fast on a straight track

### Curvilinear motion: definition.

Curvilinear motion is a type of motion in which an object moves along a curved path rather than a straight line as observed in rectilinear motion. In other words, the path in which the object moves is a curve or a combination of curves. The object’s motion involves a change of direction as well as a change of speed.

Unlike rectilinear motion, such a type of motion cannot be described using a single-coordinate axis, rather it requires to use of a two or three-coordinate axis depending on the number of dimensions involved. In curvilinear motion displacement of the object is measured using two or three coordinates. Further its velocity and acceleration can have both magnitude and displacement.

### Examples;

• A car in motion in a curved path: One common example is a car moving in a curved path.  When it moves along a curved path, it aligns with the x and y-axis. As the car drives in a curved path, that needs multiple coordinate axis to describe.
• A roller coaster is moving along a track with loops and curves.
• An athlete running on a curved track.
• A tennis ball is released from a racket in a trajectory path.
• A bicycle is moving on a curved road.
• A satellite rotating in orbit around a planet.

### Rectilinear and curvilinear motion: displacement, velocity and acceleration;

In both the rectilinear and curvilinear motion, displacement, velocity and acceleration are important factors for consideration.

#### Rectilinear and curvilinear motion: Displacement:

Displacement is the change in position of an object from its original position to its final position. In rectilinear motion displacement of an object can be measured along a single coordinate axis. However, in curvilinear motion, the scenario changes, where it requires multiple coordinate axis.

#### Equation of displacement for rectilinear motion;

The equation of displacement for rectilinear motion can be written as;

s=ut+1/2 at ˄2.

where

• s is the displacement by an object in meters(m)
• u is the initial velocity of the said object in meter/second(m/s)
• a is the acceleration of the object in meters per second squared(m/s ˄2)
• t is the time elapsed in seconds(s)

#### Equation of displacement in curvilinear motion:

The equation for curvilinear motion can be more complex than the equation for rectilinear motion. However, the displacement can be calculated using the following principle in general.

s= ʃ √(dx ˄2+dy ˄2)

Where,

• s=displacement of an object.
• dx and dy are the infinitesimal displacements in the x and y directions respectively.

Depending on the specific path, this equation needs to be solved using calculus techniques like integration.

### Rectilinear and curvilinear motion: velocity and acceleration;

Velocity is the speed of an object with direction. In other words, the rate at which the displacement of an object changes over time, whereas acceleration is the rate of change in the velocity of an object over time. In both the Rectilinear and curvilinear motion velocity and acceleration play very important roles.

In rectilinear motion, the velocity and acceleration of an object are aligned with the same direction of the motion. In such a case the velocity and acceleration can be positive, negative or even zero, which depends on the direction of the motion. However, in curvilinear motion, the velocity and acceleration of an object can have both magnitude and direction. Thus the velocity and acceleration of the object can be described by vectors.

### Rectilinear and curvilinear motion: equation of velocity

#### The equation of velocity for rectilinear motion;

Now the equation of velocity for rectilinear motion can be given as;

v=u+ at

Where,

• v=final velocity of the object in meters per second(m/s)
• u=initial velocity of the object in meters per second(m/s)
• a=acceleration of the object in meters per second squared(m/s ˄2)
• t=time elapsed in seconds(s)

This equation can be utilised to find the velocity of an object, assuming constant acceleration. If acceleration is not constant then one has to resort to calculus techniques to calculate the velocity.

#### Equation of velocity for curvilinear motion;

The  equation of velocity for curvilinear motion can be written as;

v=ds / dt,

where,

• v=velocity of an object in meters per second.(m/s)
• ds=infinitesimal displacement in the curved path in meters (m).
• dt= infinitesimal time interval in seconds(s).

The equation can be solved using a differentiated calculus technique, depending on the specific path.

### Rectilinear and curvilinear motion: equation of acceleration

#### Equation of acceleration in rectilinear motion.

The equation of acceleration in rectilinear motion can be written as;

a= (v-u) /t   where,

• a=acceleration of the object in meters per second squared(m/s ˄2)
• v=final velocity of the object in meters per second (m/s)
• u=initial velocity of the object in meters per second (m/s)
• t=time elapsed in seconds(s)

This equation represents the relationship between initial, and final velocity with time elapsed. The equation can be used to calculate the acceleration of an object in rectilinear motion, considering the uniform acceleration. If the acceleration varies then the calculus technique may be restored to measure the acceleration.

#### Equation of acceleration in curvilinear motion:

The equation of acceleration in curvilinear motion can be written as,

a=dv / dt=d ˄2s/ dt  ˄2

where,

• a=acceleration of the object in meters per second squared (m/s ˄2)
• dv= infinitesimal change in velocity over an infinitesimal time interval in meters per second squared( m/s ˄2)
• ds= infinitesimal displacement along the curved path in meters(m)
• dt= infinitesimal time interval in seconds(s)
• d ˄2s/  dt  ˄2= second derivatives of displacement with respect to time, or rate of change of velocity with respect to time.

The equation represents the rate of velocity change with respect to time or instantaneous acceleration of the object along the curved path. Calculus techniques such as differentiation or integration can be used depending on the specific path.

## Conclusion:

Rectilinear and Curvilinear Motion are two types of motion in physics, which may be distinguished on the basis of the shape of the path taken by the object. The rectilinear motion involves a straight line, whereas, the curvilinear motion takes a curved path. Rectilinear motion can be described using a single coordination axis but in the case of curvilinear motion, two or three coordination axis are needed. Displacement, velocity and acceleration are important factors in both rectilinear and curvilinear motions. However, in curvilinear motion, the velocity and acceleration can have both magnitude and direction.