# Equation of Motion (Force and Motion)

Force

Force is any influence that causes an object to undergo a certain change in its movement, direction, or geometrical construction.

A force has both magnitude and direction, making it a vector quantity.

Motion

Motion is a change in position of an object with respect to time and its reference point.

Motion is observed by attaching a frame of reference to a body and measuring its change in position relative to another reference frame.

An object’s motion cannot change unless it is acted upon by a force, as described by Newton’s first law.

Velocity

Velocity is the rate of change of the position (displacement) of an object, equivalent to a specification of its speed and direction of motion.

Speed describes only how fast an object is moving; whereas velocity gives both how fast and in what direction the object is moving.

A constant velocity means motion in a straight line (constant direction) at a constant speed.

Velocity is a vector physical quantity; both magnitude and direction are required to define it.

Acceleration

If there is a change in speed, direction, or both, then the object is said to have a changing velocity and is undergoing an acceleration.

The rate of change of velocity is “acceleration” (in m/s2), which describes how an object’s speed and direction of travel change at each point in time.

Equation of motion

The average velocity of an object moving through a displacement, d during a time interval, t is

d / t

Average velocity magnitudes are always smaller than or equal to average speed of a given particle.

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

v = u + at

The average velocity of an object undergoing constant acceleration is

(u+v) / 2

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, t, then:

(u+v) t / 2

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

x = u t + ½at²

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:

v ² = u² + 2 a x

Relative velocity

Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system.

If an object A is moving with velocity vector v and an object B with velocity vector w, then the velocity of object A relative to object B is defined as the difference of the two velocity vectors:

V (A relative to B) = v – w

Similarly the relative velocity of object B moving with velocity w, relative to object A moving with velocity v is:

V (B relative to A) = w – v