# Motion in Straight Line

## Frame of Reference

A coordinate system along with a clock makes a frame of reference. A rectangular frame of reference consists of three mutually perpendicular axes, viz. X, Y and Z. The point of intersection of three axes is called origin or O. The x, y and z coordinates of an object describe the position of the object with reference to this coordinate system. A clock is used to measure time.

**Motion:** If one or more coordinates of an object change with time, it is said that the object is in motion.

## Path Length

Total distance moved by an object with change in time is called path length. Let us choose an x-axis to understand this. In the given figure, O represents the origin. Let us assume that a car starts from O and moves to P. After that, the car moves from P to Q.

The distance covered by the car = OP + PQ

360 m + 120 m = 480 m

Here, 480 m is the path length. Path length is a scalar quantity, i.e. it has magnitude but no direction.

### Displacement

Change in position of an object is called displacement. Let us assume that x_{1} and x_{2} are positions of an object respectively at time t_{1} and t_{2}. Then displacement Δx in time Δt is given by following equation:

Δx = x_{2} - x_{1}

If x_{2} > x_{1} then Δx is positive, but if x_{2} < x_{1} then Δx is negative.

Displacement is a vector quantity, i.e. it has both magnitude and direction.

Let us go back to previous example of motion of a car. When the car moves from O to P

Then, Δx = x_{2} - x_{1}

= 360 m – 0 m = 360 m

When the car moves from P to Q

Then, Δx = x_{2} - x_{1}

= 240 m – 360 m = -120 m

The magnitude of displacement may or may not be equal to the path length. Displacement can never be greater than path length.

## Average Velocity and Average Speed

The change in position or displacement (Δx) divided by the time intervals (Δt) is called average velocity.

v = `(x_2-x_1)/(t_2-t_1)=(Δx)/(Δt)`

where, x_{2} and x_{1} are positions of object respectively at time t_{2} and t_{1}.

The SI unit of average velocity is m/s or ms^{-1}. Average velocity is a vector quantity.

This graph shows the motion of the car between t = 0 s and t = 8. Let us calculate the average velocity of car between t = 5 s and t = 7 s

v `=(x_2-x_1)/(t_2-t_1)`

`=((27.4-10.0)m)/((7-5)s)=8.7` ms^{-1}

Geometrically, the average velocity is given by the slope of straight line P_{1}P_{2}.

**Average Speed:** The total path length travelled divided by total time interval gives the average speed.

Average Speed = Total Path Length ÷ Total time interval

SI unit of average speed is same as that of average velocity.

## Instantaneous Velocity and Speed

The limit of the average velocity as the time interval Δt becomes infinitesimally small, is called instantaneous velocity at that instant.

`=(dx)/(dt)`

Instantaneous speed is simply the magnitude of velocity.

## Acceleration

The rate of change of velocity with time is called acceleration. Average acceleration over a time interval is the change of velocity divided by time interval. If v_{2} and v_{1} are instantaneolus velocities at time t_{2} and t_{1} then average acceleration is given by following equation.

a `=(v-2-v-1)/(t_2-t_1)=(Δv)/(Δt)`

The SI unit of acceleration is ms^{-2}

When a graph is plotted for velocity Vs time, average acceleration is given by the slope of straight line connecting the points corresponding to (v_{2}, t_{2} and(v_{1}, t_{1}).

Instantaneous acceleration is given by following equation:

We have read that velocity is a scalar quantity, i.e. is has both magnitude and direction. So, a change in velocity may involve change in either magnitude or direction, or a change in both. So, acceleration can result from a change in direction, change in magnitude and change in both. Velocity can be positive or negative or zero.

If the velocity of an object is v_{0} at time t = 0 and v at time t, then average acceleration is given by following equation.

a `=(v-v_0)/(t-0)`

Or, v = v_{0} + at