Instantaneous velocity and speed

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It 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 instance of time during that interval .the velocity of instance is defined as the limit of the average velocity as the time interval becomes infinitesimally small . Velocity of particle at any instant of time or at any point of its path is called instantaneous velocity.

Limiting value of Δx/ Δt as Δt aproaches zero is written as dx / dt , and is known as instantaneous velocity. Thus instantaneous velocity is:
v =lim =Δ x /Δ t

Δ->0

= dx / dt

where lim Δ -> 0 stands for the operation of taking limit as of the quantity on its right

Acceleration

Acceleration can be defined in the same way as instantaneous velocity. Acceleration is the rate of change of velocity with time. For describing average acceleration we first consider the motion of an object along X-axis.it is defined the as the rate of the change of velocity .when the velocity of object is increased it is called as positive acceleration ,when the velocity object is decrease it shows negative .

Suppose at time t1 object is at point P moving with velocity v1 and at time t2 it is at point Q and has velocity v2. Now average acceleration of object in moving from P to Q is

It will change in velocity of object with the passage of time.

Note : Acceleration can also be positive, negative and zero.