Velocity - Equation of Motion

Equation of Motion

The average velocity of an object moving through a displacement during a time interval is described by the formula:

The velocity vector v of an object that has positions x(t) at time t and x at time, can be computed as the derivative of position:

Velocity is also defined as rate of change of displacement. Average velocity magnitudes are always 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 to some point in time later .

The final velocity v of an object which starts with velocity u and then accelerates at constant acceleration a for a period of time is:

The average velocity of an object undergoing constant acceleration is, where u is the initial velocity and v is the final velocity. To find the position, x, of such an accelerating object during a time interval, then:

When only the object's initial velocity is known, the expression,

can be used.

This can be expanded to give the position at any time t in the following way:

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:

The above equations are valid for both Newtonian mechanics and special relativity. Where Newtonian mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in Newtonian mechanics, all observers agree on the value of t and the transformation rules for position create a situation in which all non-accelerating observers would describe the acceleration of an object with the same values. Neither is true for special relativity. In other words only relative velocity can be calculated.

In Newtonian mechanics, the kinetic energy (energy of motion), of a moving object is linear with both its mass and the square of its velocity:

The kinetic energy is a scalar quantity.

Escape velocity is the minimum velocity a body must have in order to escape from the gravitational field of the earth. To escape from the Earth's gravitational field an object must have greater kinetic energy than its gravitational potential energy. The value of the escape velocity from the Earth's surface is approximately 11 100 m/s.