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.
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.
Change in position of an object is called displacement. Let us assume that x1 and x2 are positions of an object respectively at time t1 and t2. Then displacement Δx in time Δt is given by following equation:
Δx = x2 - x1
If x2 > x1 then Δx is positive, but if x2 < x1 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 = x2 - x1
= 360 m – 0 m = 360 m
When the car moves from P to Q
Then, Δx = x2 - x1
= 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, x2 and x1 are positions of object respectively at time t2 and t1.
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
Geometrically, the average velocity is given by the slope of straight line P1P2.
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.
Instantaneous speed is simply the magnitude of velocity.
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 v2 and v1 are instantaneolus velocities at time t2 and t1 then average acceleration is given by following equation.
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 (v2, t2 and(v1, t1).
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 v0 at time t = 0 and v at time t, then average acceleration is given by following equation.
Or, v = v0 + at