# Rectilinear Motion

Rectilinear Motion (or Linear Motion) is a specific form of motion. It is a motion that occurs in a single direction or axis. All the motion-related parameters are along one direction. Hence, the motion is also called motion in one dimension

Index

## Rectilinear Motion Definition

Rectilinear motion definition can be stated as a motion that occurs along a single direction only. It can also be called straight-line motion. The entire motion can be described without needing to use vectors.

## Rectilinear Motion Examples

Here are some of the rectilinear motion examples from our daily life.

• An athlete running on a track
• A bead moving on a fixed rod
• A train moving on a straight track
• Swimmer swimming in parallel
• An object falling straight down

## Quantities Involved

Here we discuss some quantities involved in linear motion.

### Distance

It is the total distance covered by the moving object/particle. For example, if an object moves 10 meters forward and 6 meters backwards, the total distance covered is 10+6 = 16 meters.

### Displacement

It is the NET distance between starting and endpoint of motion. Using the same example as above, if the body moves 10 m forwards and 6 m backwards, it is only (10-6) m away from starting point. Thus, displacement is just 4 meters.

### Velocity

Velocity $$(\mathbf{v})$$ is the rate of change of displacement. It has a direction. In linear motion, this direction is represented by a positive or negative sign. The boldface indicates the fact that it has a direction.

#### Average Velocity

It is given by the displacement $$\mathbf{s_2 – s_1}$$ divided by the total interval of time $$(t_2 – t_1)$$ it takes cover.

In other words,

$$\bar{\mathbf{v}} = \frac{\mathbf{s_2-s_1}}{t_2 – t_1}$$

#### Instantaneous Velocity

It is the limit of average velocity when time interval goes to zero. It gives the velocity of an object at a specific instant.

$$\mathbf{v} = \lim_{{(t_2 – t_1)} \rightarrow 0} \frac{\mathbf{s_2-s_1}}{t_2-t_1} = \lim_{{\Delta t} \rightarrow 0} \frac{\mathbf{\Delta s}}{\Delta t} = \frac{d\mathbf{s}}{dt}$$

Thus, it is the time-derivative of displacement.

### Speed

Speed ($$v)$$ is the magnitude of velocity. As such, it has no direction and is always non-negative. Unlike velocity, it is denoted without boldface as it has no direction.

#### Average Speed

It is denoted by $$\bar{v}$$. It can be given by distance ($$Delta d)$$ covered in a given time interval $$(\Delta t)$$.

Mathematically,

$$\bar{v} = \frac{\Delta d}{\Delta t}$$

### Acceleration

It is the rate of change of velocity. It is denoted by $$\mathbf{a}$$. The boldface denotes that it has direction. In linear motion, the direction is given by the sign.

Mathematically,

$$\mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2 \mathbf{s}}{dt^2}$$

## Types of Rectilinear Motion

Now, let us discuss different types of rectilinear motion we encounter on daily basis.

### Uniform Rectilinear Motion

In this type of linear motion, velocity is constant. There is no acceleration and no net external force acting on the object.

### Uniformly Accelerated Rectilinear Motion

In this linear motion, velocity is changing but the acceleration is constant also known as Kinematic Equations. The motion of the body an be given by the three equations of motion

$$\mathbf{v} = \mathbf{u} + \mathbf{a}t$$

$$\mathbf{s} = \mathbf{u}t + \frac{1}{2} \mathbf{a} t^2$$

$$\mathbf{v^2} – \mathbf{u^2} = 2 \mathbf{a} \cdot \mathbf{s}$$

Know more on Rectilinear Motion Formula

### Non-uniformly Accelerated Rectilinear Motion

Here, neither velocity nor acceleration is constant. Both can vary with time.

## FAQs

What is rectilinear motion?

It is a type of motion that happens along one direction or axis only. It is also known as linear motion.

What is the difference between velocity and speed?

Velocity is the rate of change of displacement, it has both magnitude and direction, while speed is simply the magnitude of velocity, it doesn’t have direction.

What are the three equations of motion?

The three equations of motion are:
$$\mathbf{v} = \mathbf{u} + \mathbf{a}t$$
$$\mathbf{s} = \mathbf{u}t + \frac{1}{2} \mathbf{a} t^2$$
$$v^2 – u^2 = 2 \mathbf{a} \cdot \mathbf{s}$$
They are applicable for uniformly accelerated motion.

What is the difference between rectilinear and curvilinear motion?

Rectilinear motion occurs in a single direction, and is thus one-dimensional. It can be called motion in a straight line.
Curvilinear motion occurs along a curve rather than a line. It can thus be two- or three-dimensional.

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