MOTION IN A STRAIGHT LINE

 FRAME OF REFERENCE

Various physical phenomena, events, etc., require position and time coordinates to specify their complete description. Since nothing seems stationary in the universe, some arbitrary reference systems have to be defined with respect to which the position and the time coordinates for the event, etc., can be assigned. Such a system is called a 'frame of reference'.

INERTIAL AND NON-INERTIAL REFERENCE FRAMES

Newton's first law tells us that we can find a reference frame relative to which a body remains at rest or in uniform motion along a straight line when no net external force acts upon it. Such a reference frame is called an 'inertial reference frame'. Thus an inertial frame is one in which Newton's first law correctly describes the motion of a body not acted upon by a net force. Such frames are either at rest or moving with uniform velocity with respect to distant stars. Thus, they are unaccelerated and non-rotating.

A reference frame attached to the earth can be considered as an inertial frame for almost all laboratory-scale terrestrial experiments, although it is not precisely inertial because of the axial and orbital motions of the earth. But if a reference frame is inertial, then every other reference frame which is in uniform motion relative to it is also an inertial frame. Thus, an aircraft in steady flight over the earth is just as suitable an inertial frame as the earth itself.

Relationship between the inertial and non-inertial reference frames.

On the other hand, a reference frame that is accelerated or rotating is called a 'non-inertial' reference frame. For example, a referential frame attached to an aircraft that is making its take-off (accelerated) run is a non-inertial frame. Similarly, a rotating merry-go-round is also a non-inertial frame.

CONCEPT OF POINT MASS

The point mass is also called point object or simply the particle. An object is said to be a point object or a point mass when the change in its position is very large as compared to its own size. For example

• A person standing on the platform is regarded as the point object for the train passing through the platform.
• Earth is considered a point object since for motion around the sun, the size of the earth (diameter) is very small as compared to the distance between the sun and the earth.

MOTION AND REST

An object is said to be in motion if its position changes with respect to its surroundings in a given time. On the other hand, if the position of the object does not change with respect to its surroundings, it is said to be at rest.

Motion involves a change in position

To study the motion of an object, one has to study the change in the position of the object with respect to its surroundings. The position of an object in space is specified by three coordinates x,y, and z. The position of the object changes due to a change in one or two or all three coordinates.

• The motion of an object is said to be one dimensional when one of the three coordinates specifying the position of the object changes with time. The motion of a car on the road, the motion of a train along a railway train, or an object falling freely under gravity are examples of one-dimensional motion.
• The motion of an object is said to be two-dimensional when two of the three coordinates specifying the position of the object change with time. The motion of a planet around the sun, a body revolving in a circle and an insect crawling on a floor are examples of motion in two dimensions.
• The motion of an object is said to be three-dimensional when all the three coordinates specifying the position of the object change with time. The motion of gas molecules, the motion of a bird in the sky, or a flying kite are examples of motion in three dimensions.

DISTANCE AND DISPLACEMENT

The position of the moving object goes on changing with respect to time. The length of the actual path covered by a body in a time interval is called 'distance'; while the difference between the final and the initial positions of an object is called 'displacement'.

If the initial position of an object with respect to the reference point is s1 and after some time it changes to s2, then the magnitude of displacement of the object is s2-s1. If the value of  s2 is greater than s1 then the displacement is positive, but if the value of s2 is less than s1 then the displacement is negative. If s2=s1 then the displacement is zero.

There is an important difference between 'distance' and 'displacement'. Distance is a scalar quantity which has magnitude only. Displacement is a vector quantity which has both magnitude and direction.

 SPEED 

The distance travelled by an object in unit interval of time is called the 'speed' of the object. 

Speed is represented  by v. Its unit is metre/second (ms^-1) and its dimensions are [LT^-1]. 

If the object is covering equal distances in equal time-intervals, then its speed is said to be 'uniform'. If, however, the object is covering different distances in equal time-intervals, then its speed is said to be 'variable'. In this case, the average speed of the object is determined by dividing the total distance travelled by the total time. If the motion of an object is uniform then its speed and average speed are identical.

If the speed of an object is continuously changing with time, then its speed is at a particular instant is called its 'instantaneous speed'. The speedometer in a car indicates the instantaneous speed of the car.

Speed can be thought of as the rate at which an object covers distance. A fast-moving object has a high speed and covers a relatively large distance in a given amount of time, while a slow-moving object covers a relatively small amount of distance in the same amount of time.

VELOCITY

The displacement of an object in unit interval of time, is called the 'velocity' of the object:

The velocity is also represented by v and its unit (ms^-1) and dimensions [LT^-1] are the same as those of speed.

Suppose an object is moving along a straight line. Suppose, with respect to some reference point, its position is s1 a time t1 and becomes s2 at time t2. It means that in time interval (t2-t1) the displacement of hte object is (s2-s1). hence, the average velocity of the object during this time-interval would be

For writing the difference in a quantity we use the symbol Δ (delta). We can write Δs for the difference in positions (s2-s1) and Δt for the difference in times (t2-t1). Using the symbol Δ, the average velocity of the object is given by

When Δt becomes infinitesimally small(Δt→ 0) , then the above formula  we shall be knowing the velocity of the object at a particular instant of time. This velocity is the 'instantaneous velocity' of the object and is given by

There is difference between 'speed' and 'velocity'. Speed is a scalar quantity. It has magnitude only, no direction. Velocity is a vector quantity. It includes magnitude as well as direction. 

If, either the magnitude only or the direction only is changed, even then the velocity will be changed. If two objects are travelling in different directions with the same speed, then their velocities will not be same.

ACCELERATION

If the velocity of a moving object is changing, then its motion is called 'accelerated motion'. The change is velocity maybe due to change in magnitude or in direction or in both. If the body is moving along a straight line , then only the magnitude of velocity(speed) changes.

The time-rate change of velocity of an object is called 'acceleration' of that object.

The average acceleration of the object in time interval t2-t1 is

If the time-interval  Δt is infinitesimally small(Δt→ 0), then that above formula gives acceleration 'at a particular time'. This is called 'instantaneous acceleration' and is given by 

The unit of acceleration is metre/second² (ms^-2) and its dimensional formula is [LT^-2]. 

If the velocity of an object undergoes equal changes in equal interval of times, then its acceleration is said to be 'uniform'. If the magnitude of th velocity of an object is increasing with time, then the acceleration of the object is positive. If the magnitude of the velocity is decreasing, then the acceleration is negative and its then called 'retardation'.