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Consider position – time plot as shown below showing a trip by a motor car.
Motion in straight line
Determine :
Characteristics of motion : One dimensional, variable speed
(i) Total distance in the round trip = 120 + 120 = 240 Km
(ii) Displacement = 120 – 120 = 0
(iii) Average speed for the round trip
(iv) Average speed during motion from O to C
As magnitude of average velocity is positive, the direction of velocity is in the positive x – direction. It is important to note for motion in one dimension and in one direction (unidirectional), distance is equal to the magnitude of displacement and average speed is equal to the magnitude of average velocity. Such is the case for this portion of motion.
(v) The part of motion for which average velocity is equal in each direction.
By inspection of the plots, we see that time interval is same for motion from B to C and from C to B (on return). Also, the displacements in these two segments of motion are equal. Hence magnitudes of velocities in two segments are equal.
Motion in straight line
(vi) Compare speeds in the portion OB and BD.
By inspection of the plots, we see that the motor car travels equal distances of 60 m. We see that distances in each direction is covered in equal times i.e. 2 hrs. But, the car actually stops for 1 hour in the forward journey and as such average speed is smaller in this case.
Instantaneous velocity is defined exactly like speed. It is equal to the ratio of total displacement and time interval, but with one qualification that time interval is extremely (infinitesimally) small. Thus, instantaneous velocity can be termed as the average velocity at a particular instant of time when Δt tends to zero and may have entirely different value than that of average velocity. Mathematically,
As Δt tends to zero, the ratio defining velocity becomes finite and equals to the first derivative of the position vector. The velocity at the moment ‘t’ is called the instantaneous velocity or simply velocity at time ‘t’.
Position - time plot provides for calculation of the magnitude of velocity, which is equal to speed. The discussion of position – time plot in the context of velocity, however, differs in one important respect that we can also estimate the direction of motion.
Instantaneous velocity
In the figure above, as we proceed from point B to A through intermediate points B’ and B’’, the time interval becomes smaller and smaller and the chord becomes tangent to the curve at point A as Δ t → 0. The magnitude of instantaneous velocity (speed) at A is given by the slope of the curve.
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