Motion in a Straight Line

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Distance

During motion in a straight line, the point object occupies a definite position on the path at each instant. Therefore, to describe the motion, one should specify the length of the path covered by the point object and the instant of time. The length of path covered by the object is the distance 's'. A knowledge of the length of the path or distance covered indicates the actual path taken by the object during its journey. For example, if A and B represent two positions and an ant is able to reach B along paths 1 and 2, then length of path 1 is different from length of path 2.

Distance is a scalar quantity.

Displacement

The displacement of a moving particle is its change of position in a particular direction.

To know the displacement of a moving particle, we must know both the length and the direction of the line joining the two positions of the moving particle. Hence, the displacement of a particle or an object involves both magnitude and direction.

For example, A man walks 3 km due east and then 4 km due north. This is illustrated in the figure above. The man walks from A to B due east and then from B to C due north. However, his total displacement is from A to C, which is 5 km at an angle of tan^-1 4/3 north of east.

It is important to remember that displacement is a vector - it has both magnitude and direction. Moreover, the displacement between two points has a unique value.

The displacement of an object between two points does not give us any information about the type of motion or path followed by the object between those two points. For example, the displacement of the 2 ants in the above example, are same, although their distances are different.

SI unit of both distance and displacement is metre (M) and dimensions are [M⁰L¹T⁰].

Speed

Definition

The speed of a moving particle is the rate at which it describes its path.

Mathematically, this may be expressed in the following way.

If a moving particle covers a distance s in a given interval of time, say, t then, the quantity s/t, is the measure of the speed of the moving particle at the instant under consideration.

a) A particle is said to be moving with uniform speed when it moves through equal lengths of its path in equal intervals of time, however small these time intervals may be.

If a particle is moving with speed u, then in each unit of time, it moves u units of length. Hence, in t units of time, it moves a distance s = u.t.

b) An object is said to be moving with a variable speed if it covers equal distances in unequal intervals of time or unequal distances in equal intervals of time, however small these intervals may be.

c) When an object is moving with a variable speed, then the average speed for the given motion is defined as the ratio of the total distance travelled by the object to the total time taken is

d) If an object has a variable speed such that the speed varies at different instants, then the speed at any instant is called the instantaneous speed. If at an instant t, an object covers a distance Ds in a small interval of time Dt so that Dt g 0 [i.e., Dt approaches zero] then,

The speed is a scalar quantity and gives no idea about the direction of the motion of the object. The speed of an object (like distance) can be zero or positive but never negative.

Velocity

Definition

The velocity of a moving particle is the rate of its displacement or the rate of change of its displacement with time.

Therefore, velocity possesses both magnitude and direction and hence, is a vector.

A particle is said to be moving with uniform velocity when it is moving along a straight line or in a constant direction and covering equal distances in equal intervals of time, however small these intervals may be.

Uniform velocity of a moving object is measured by its displacement per unit time.

Variable velocity is measured at any instant by the displacement that the moving object would have in unit time. In other words, we would have to calculate the instantaneous velocity.

The SI unit of speed (and velocity) is metre/second [i.e., m/s] and dimensions are [M⁰L¹T^-1].

Example

A train moves with a velocity 40 km/hr for the first half hour, 60 km/hr in the next one hour and 20 km/hr in the last half hour. The distances covered in these time intervals are

S₂ = 60 km

Therefore, the total distance covered by the train is 90 km. Since the distance has been covered in an interval of 2  hours, the average