Motion in a Straight Line

Introduction

Motion is common to several things that move around us. Falling leaves, walking, running all have motion common in them. Also, our earth is in motion though we don’t feel it. Motion can be in a straight line (1D), in a plane (2D), or space (3D). The motion of an object in one direction is called motion in a straight line. For example, the motion of a train on straight rails, motion of a free-falling body, an athlete running on a straight track, etc.

A change in the position of an object with respect to time is called motion. Kinematics and dynamics are the two branches of physics that detail the motion of an object. Kinematics describes objects occurring without the cause of the motion whereas the dynamics relate the motion of objects to the forces that cause them.

Point object- Those objects that cover a large length in comparison to the size of the objects, then objects are considered as point objects.

Reference System- Motion of an object is always considered with respect to a reference system. A reference system is made by having arbitrary points as the origin and imagining a coordinate system for it.

Types of motion

The motion of an object is described only by its position. In some cases, to determine the position of an object, we use a 3-coordinate system, or sometimes 1- or 2- the coordinate system is required. On this basis, motion is categorized into:

(1-D), or one-dimensional motion- in this dimension, the particle moves in a straight line. For example- a freely falling body under gravity, the motion of a train in a straight rail track, etc.
(2-D), or two-dimensional motion- Object moving in a plane is said to undergo two-dimensional motion. For example- motion of bullets, carrom board coins, etc.
(3-D), or three-dimensional motion- Objects moving in space are said to undergo three-dimensional motion—for example- the motion of a kite, the motion of an airplane, etc.

Speed And Velocity

Distance is the total path length covered by an object or a body traveled by it.

But, displacement is the difference between the initial and final position vector of a body. It is the shortest distance between the two positions and has a definite direction. Therefore, displacement is the vector quantity, and the distance is the scalar quantity. Both distance and displacement are measured in m or km. their dimension is [L].

Factors distinguishing between displacement and distance

Displacement has direction, whereas distance has no direction.
The magnitude of displacement can be both positive or negative or zero, whereas distance is always positive.
Distance is a scalar quantity and displacement is a vector quantity.

Speed is the rate of change of distance with respect to time, whereas the rate of change of displacement is known as velocity. Velocity is a vector quantity and speed is a scalar quantity. Speed and velocity are expressed in m/s. The dimensional formula is [LT-1]. Velocity can be changed by change in magnitude or by change in direction or both.

Uniform Speed and Uniform Velocity

Uniform speed – When an object is moving with uniform speed and covering equal distance in equal interval of time, then the object is said to have uniform speed.

Uniform velocity – When an object is moving with uniform velocity covering equal displacements in equal intervals of time, then the object is said to have uniform velocity.

Variable Speed and Variable Velocity

Variable Speed- An object covering unequal distance in equal intervals of time is said to have variable speed.

Variable Velocity- An object covering unequal displacements in equal intervals of time is said to have variable velocity.

Average Speed and Average velocity

Average Speed is the ratio of total path length covered in given interval of time.

Mathematically,

Average speed = Total distance covered

Total time taken

vav=xt ……………. (1)

Average velocity is the distance traveled by an object with different velocities. It is the displacement per unit time. Let x1 and x2 be positions of an object at t1 and t2, respectively.

Mathematically,

vav= displacementtime

vav = xt ………equation (2)

Here, x denotes a change in position and t is a change in time. Here, the bar above the symbol for velocity indicates average quantity..

The average velocity of a body is obtained by dividing the total distance traveled by the total time taken:

When the motion of an object is in the same direction along a straight line then the magnitude of the average speed is the same as the magnitude of the average velocity. But it is not always the same.

Example 1- The position of an object moving along the x-axis is defined as x = 20t2, where t is the time measured in seconds and position is expressed in metres. Calculate the average velocity of the object over the time interval from 4s to 5s.

Solution: Given,

x = 20t2

We know that the average velocity is given by the relation

vav = xt

At t1 = 4s,

x1 = 20 × (4)2

= 20 × 16 = 320 m

Similarly, for t = 5s

x2 = 20 × (5)2

= 20 × 25 = 500 m

vav = xt = (500–320)m / (5-4)s = 180m / 1s = 180 m/s

Hence, average velocity = 180 ms–1.

Example 2- A person runs on a 600m circular track and comes back to the starting point in the 300s. Calculate the average speed and average velocity.

Solution- Given,

Total length of the track = 600m.

Time taken to cover this length = 300s

Hence,

average speed = total distance traveled

time taken

= 600 ms–1

300

= 2 ms–1

As the person comes back to the same point, the displacement is zero. Therefore,

The average velocity is also zero.

Instantaneous Speed and Instantaneous Velocity

Instantaneous Speed- It is the instant speed of an object

Instantaneous Velocity- it is the instant velocity of an object at any point of its path.

Relative velocity

Relative velocity is the rate of change of the relative position of an object with respect to the other object is called the relative velocity of that object with respect to the other.

Example 3- A train A is moving on a straight track (or railway line) from East to West with a speed of 90km h–1. Another train B is moving from West to East with a speed of 100km h–1. What is the velocity of B relative to train A?

Solution- Considering the direction from South to North as positive, we have

velocity (vB) of train B = + 100km h–1

and, velocity (vA) of train A = – 90km h–1

Hence, the velocity of train B relative to train A

= vB – vA

= 100 – (– 90) = 190km h–1.

Here, the velocity of one train with respect to the other train is equal to the sum of their respective velocities. Train moving in the opposite direction appears to move very fast, whereas a train moving in the same direction appears to be very slow.

Acceleration

Sometimes the speed of an object, while traveling in a car or bus, becomes fast or sometimes slows down, indicating a change in velocity with respect to time. As we know, velocity is the rate of change in the displacement with respect to time, whereas acceleration is the rate of change of velocity with respect to time. Acceleration is the vector quantity. SI unit of acceleration is ms-2.

Mathematically,

aav= v2-v1t2-t1=vt

In 1-D motion, vector notation is not used for acceleration. In 1-D motion, the acceleration is positive when the acceleration is in the same direction of motion. When acceleration is negative, then it is called deceleration or retardation.

Example 4- The velocity of a car moving towards the East increases from 0 to 16ms–1 in 4.0 s. Calculate its average acceleration.

Solution- Given,

v1 = 0 m s–1

v2 = 16 m s–1

t = 4.0 s

a = (16.0m s–1)

4.0s

= 4.0 m s–2.

Uniform Acceleration of an object undergoes equal changes in velocity in equal intervals of time.

Average Acceleration is a change in velocity divided by the time taken by an object during the interval.

Instantaneous Acceleration is defined as the limit of average acceleration as the time (t)goes to zero.

On the basis of nature of movement, motion is classified as follows:

Linear Motion
Rotary Motion
Oscillatory Motion

Linear Motion

In this motion, particle moves from one point to another point either in a straight line or on a curved path. On the basis of path of motion, linear motion is categorized into-

Rectilinear motion- motion in a straight line.
Curvilinear motion- motion in a curved line.

Types of Linear Motion

Linear motion is also called rectilinear motion. It is of two types-

Uniform linear motion having constant velocity or zero acceleration
Non- Uniform linear motion having variable velocity or non-zero acceleration

Uniform Motion in a Straight Line

When an object travels in a straight line covering an equal distance in an equal interval of time it is said to have uniform motion in a straight line. In other words, an object has uniform acceleration while its rate of change of velocity remains constant.

For example- If a bike travels at a speed of 70km/hr, then it will cover 1km/hr. Here, the motion of bike acceleration is uniform.

Non-Uniform Motion in a Straight Line

When the velocity of an object changes unequally in equal intervals of time, then the rate of change of velocity changes at different intervals of time.

For example- A boy kicking a ball. It may cover a distance of 2m the first time, 4m the second time, 6m the third time, and so on.

Rotary motion

It is a type of motion in which an object moves on its own axis—for example- the spinning of a wheel.

Oscillatory motion

It is a type of motion in which a body moves around its mean position. For example- the pendulum of the clock is an example of oscillatory motion.