2.2 Rectilinear motion

Consider an object moving along a straight line.

Let us assume this line to be along the x-axis.

Let x 1 and x 2 be the position vectors of the body at times t1 and t2 during its motion.

The following quantities can be defined for the motion.

1. Displacement:

The displacement of the object between t1 and t2 is the difference between the position vectors of the object at the two instances.

Its direction is along the line of motion of the object.

Its dimensions are that of length.

For example, if an object has travelled through 1 m from time t1 to t2 along the +ve x-direction, the magnitude of its displacement is 1 m and its direction is along the +ve x-axis.

On the other hand, if the object travelled along the +ve y direction through the same distance in the same time, the magnitude of its displacement is the same as before, i.e., 1 m but the direction of the displacement is along the +ve y-axis.

2. Path length:

This is the actual distance travelled by the object during its motion.

It is a scalar quantity and its dimensions are also that of length.

If an object travels along the x-axis from x = 2 m to x = 5 m then the distance travelled is 3 m.

In this case the displacement is also 3 m and its direction is along the +ve x-axis.

However, if the object now comes back to x = 4, then the distance through which the object has moved increases to 3 + 1 = 4 m.

Its initial position was x = 2 m and the final position is now x = 4 m and thus, its displacement is x = 4 – 2 = 2 m, i.e., the magnitude of the displacement is 2 m and its direction is along the +ve x-axis.

If the object now moves to x =1, then the distance travelled, i.e., the path length increases to 4 + 3 =7 m while the magnitude of displacement becomes 2 – 1 = 1 m and its direction is along the negative x-axis.

3. Average velocity:

This is defined as the displacement of the object during the time interval over which average velocity is being calculated, divided by that time interval.

As displacement is a vector quantity, the velocity is also a vector quantity.

Its dimensions are [L^1 M^0 T^-1].

4. Average speed:

This is defined as the total path length travelled during the time interval over which average speed is being calculated, divided by that time interval.

It is a scalar quantity and has the same dimensions as that of velocity, i.e., [L^1 M^0 T^-1].

If the rectilinear motion of the object is only in one direction along a line, then the magnitude of its displacement will be equal to the distance travelled and so the magnitude of average velocity will be equal to the average speed.

However if the object reverses its direction (the motion remaining along the same line) then the magnitude of displacement will be smaller than the path length and the average speed will be larger than the magnitude of average velocity.

5. Instantaneous velocity:

Instantaneous velocity of an object is its velocity at a given instant of time.

It is defined as the limiting value of the average velocity of the object over a small time interval (t) around t when the value of the time interval (t) goes to zero.

6. Instantaneous speed:

Instantaneous speed is the speed of an object at a given instant of time t.

It is the limiting value of the average speed of the object taken over a small time interval (t) around t when the time interval goes to zero.

In such a limit, the path length will be equal to the magnitude of the displacement and so the instantaneous speed will always be equal to the magnitude of the instantaneous velocity of the object.

7. Acceleration:

Acceleration is defined as the rate of change of velocity with time.

It is a vector quantity and its dimensions are [L^1 M^0 T^-2].