Explain instantaneous velocity and discuss how it can be determined from `x to t` graph.

Explanation

The average velocity tells us how fast an object has been moving over a given time interval but does not tell us how fast it moves at different instants of time during that interval. For this, we define instantaneous velocity.
The velocity at an instant is defined as the limit of the average velocity as the time interval At becomes infinitesimally small. In other words,
`v = lim _(Delta t - to 0) (Delta x)/(Delta t )`
`v = (dx)/(dt) =x`
In the language of calculus it is the differential coefficient of x with respect to t and is denoted by `(dx)/(dt).`
We can obtain the value of velocity at an instant either graphically or numerically.(1) Graphical Method :
Suppose the below given x t graph is for non-uniform motion of the car and we want to obtain graphically the value of velocity at time t = 4 s.
For this case, the variation of velocity with time is found to be as shown in figure.
Suppose, by keeping t = 4s in mind, if we take different time intervals `Deltat _(1), Delta t_(2), Delta t _(3),......` then for these time intervals displacement will be `P_(1) P_(2), Q_(1) Q_(2), T_(1) T_(2),...` and corresponding displacement are `Delta x_(1), Delta x _(2) Delta x _(3),.....` .
In all these time intervals the average velocity comes nearer to some exact value and when the time interval becomes zero it means`(Delta t lim _(x to 0)` at that time average velocity gives instantaneous velocity at t = 4 s. It is difficult to use graphical method always.
Slope of the graph is instantaneous velocity at that point (Slope `= tantheta`).
For the motion in straight line the slope is construct. Hence, average and instantaneous velocities are same.(2) Numerical Method: For the graph shown in figure
` x = 0.08 t ^(3).` Table gives the value of Ax/At calculated for at 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) = (t - (Delta t)/(2)) and t _(2) = (t + (Delta t )/(2))` and the fourth and the fifth columns give the corresponding values of x.
`i.e. x(t_(1)) = 0.08 t_(1) ^(3) and x(t_(2)) = 0.08 t_(2)_(3)`
The sixth column lists the difference `Delta x = x(t_(2)) - x(t_(1))` and the last column gives the ratio of Ar and `Deltat,` i.e. the average velocity corresponding to the value of at listed in the first column.
As the time interval becomes smaller, the velocity at t = 4s can be obtained.