example, "5 metres per second" is a scalar and not a vector, whereas "5 metres per second east" is a vector. The average velocity v of an object moving through a displacement ( \Delta \mathbf{d}) during a time interval ( \Delta t) is described by the formula:

    \mathbf{\bar{v}} = \frac{\Delta \mathbf{d}}{\Delta t}.

The rate of change

he velocity vector v of an object that has positions x(t) at time t and x(t + \Delta t) at time t + \Delta t, can be computed as the derivative of position:

    \mathbf{v} = \lim_{\Delta t \to 0}{{\mathbf{x}(t+\Delta t)-\mathbf{x}(t)} \over \Delta t}={\mathrm{d}\mathbf{x} \over \mathrm{d}t}.

ays smaller than or equal to average speed of a given particle. Instantaneous velocity is always tangential to trajectory. Slope of tangent of position or displacement time graph is instantaneous velocity and its slope of chord is average velocity.

The equation for an object's velocity can be obtained mathematically by evaluating the integral of the equation for its acceleration beginning from some initial period time t_0 to some point in time later t_n.

The final velocity v of an object which starts with velocity u and then accelerates at constant acceleration a for a period of time \Delta t is:

    \mathbf{v} = \mathbf{u} + \mathbf{a} \Delta t. 

This can be expanded to give the position at any time t in the following way:

    \mathbf{x}(t) = \mathbf{x}(0) + \Delta \mathbf{x} = \mathbf{x}(0) + \mathbf{u} \Delta t + \frac{1}{2}\mathbf{a} \Delta t^2,

These basic equations for final velocity and position can be combined to form an equation that is independent of time, also known as Torricelli's equation:

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