Motion in a Straight Line

Motion is common to everything in the universe. We walk, run and ride a bicycle. Even when we are sleeping, air moves into and out of our lungs and blood flows in arteries and veins. We see leaves falling from trees and water flowing down a dam. Automobiles and planes carry people from one place to the other. The earth rotates once every twenty-four hours and revolves round the sun once in a year. The Sun itself is in motion in the Milky Way, which is again moving within its local group of galaxies. Motion is change in position of an object with time. How does the position change with time ? In this chapter, we shall learn how to describe motion. For this, we develop the concepts of velocity and acceleration. We shall confine ourselves to the study of motion of objects along a straight line, also known as rectilinear motion. For the case of rectilinear motion with uniform acceleration, a set of simple equations can be obtained. Finally, to understand the relative nature of motion, we introduce the concept of relative velocity. In our discussions, we shall treat the objects in motion as point objects. This approximation is valid so far as the size of the object is much smaller than the distance it moves in a reasonable duration of time. In a good number of situations in real-life, the size of objects can be neglected and they can be considered as point-like objects without much error. In Kinematics, we study ways to describe motion without going into the causes of motion. What causes motion described in this chapter and the next chapter forms the subject matter of Chapter 4.

Summary Points to ponder Exercises

$$v=\underset{\Delta t\to 0}{\lim }\frac{\Delta x}{\Delta t}$$

$$=\frac{dx}{dt}$$

where the symbol ${\lim }_{\Delta t\to 0}$ stands for the operation of taking limit as $\Delta t\to 0$ of the quantity on its right. In the language of calculus, the quantity on the right hand side of Eq. (2.1a) is the differential coefficient of $x$ with respect to $t$ and

It is the rate of change of

position with respect to time, at that instant. We can use Eq. (2.1a) for obtaining the value of velocity at an instant either graphically or numerically. Suppose that we want to obtain graphically the value of velocity at time $t=4$ s (point P) for the motion of the car represented in Fig. 2.1 calculation. Let us take $\Delta t=2$ s centred at $t=4$ s. Then, by the definition of the average velocity, the slope of line ${\mathrm{P}}_1{\mathrm{P}}_2$ (Fig. 2.1) gives the value of average velocity over the interval 3 s to 5 s.

Now, we decrease the value of $\Delta t$ from 2 s to 1 s. Then line ${\mathrm{P}}_1{\mathrm{P}}_2$ becomes ${\mathrm{Q}}_1{\mathrm{Q}}_2$ and its slope gives the value of the average velocity over the interval 3.5 s to 4.5 s. In the limit $\Delta t\to 0$, the line ${\mathrm{P}}_1{\mathrm{P}}_2$ becomes tangent to the position-time curve at the point P and the velocity at $t=4$ s is given by the slope of the tangent at that point. It is difficult to show this process graphically. But if we use numerical method to obtain the value of the velocity, the meaning of the limiting process becomes clear. For the graph shown in Fig. 2.1, $x=0.08t^3$. Table 2.1 gives the value of $\Delta x/\Delta t$ calculated for $\Delta t$ equal to 2.0 s, 1.0 s, 0.5 s, 0.1 s and 0.01 s centred at $t=4.0$ s. The second and third columns give the

value of $t_1=\left(t-\frac{\Delta t}{2}\right)$

and $t_2=\left(t+\frac{\Delta t}{2}\right)$

and the

fourth and the fifth columns give the

$t=4.0$ s, i.e. the value of $\frac{dx}{dt}$

at $t=4.0$ s. In this