# Instantaneous velocity and speed.

Instantaneous velocity:

Beside instantaneous velocity , average velocity is a way of defining how fast a body is moving.But  average velocity is only the average of the velocity of body over a given time interval ,  and not the exact measure of velocity of a body at an instant.

The exact measure of velocity of a body at an instant or over a very-very-very short time interval $\Delta t$ is known as instantaneous velocity.

Mathematically instantaneous velocity of an object is the limiting value  of average velocity as $\Delta t$ tends towards 0.

Or:

Instantaneous velocity = $\displaystyle\lim_{\Delta t\to 0}\frac{\Delta x}{\Delta t}$

And instantaneous velocity is also the derivative of the “position of the object” ( “x” ) with respect to “time interval”( “t” )

or, Instantaneous velocity = $\frac{d}{d t} x$

In a graph of motion of an object drawn “x” versus “t”; Instantaneous velocity of the object is the slope of tangent line drawn at the position representing that instant.

Instantaneous velocity is a vector quantity and thus have  it’s direction ( positive or negative ) associated with it along with it’s magnitude.

Speed:

Speed of an object is the magnitude of the vector quantity “Instantaneous velocity” of the object.

Or , speed is similar to Instantaneous velocity and is different only because speed is a scalar quantity and doesn’t deals with the direction of the movement.

For example:

1. If the Instantaneous velocity of an object is  -7m/s then the speed of the object at that instance is 7m/s.

2. If the Instantaneous velocity of an object is  +19m/s then the speed of the object at that instance is 19m/s.

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