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So to be more precise, this is the

In Newtonian mechanics, an equation of motion M takes the general form of a 2nd order ordinary differential equation (ODE) in the position r (see below for details) of the object:

where t is time, and each overdot denotes a time derivative.
The initial conditions are given by the constant values at t = 0:

An alternative dynamical variable to r is the momentum p of the object (though less commonly used), i.e. a 2nd order ODE in p: with initial conditions (again constant values)

The solution r (or p) to the equation of motion, combined with the initial values, describes of the system for all times after t = 0. For more than one particle, there are separate equations for each (this is contrary to a statistical ensemble of many particles in statistical mechanics, and a many-particle system in quantum mechanics - where all particles are described by a single probability distribution). Sometimes, the equation will be linear and can be solved exactly. However in general, the equation is non-linear, and may lead to chaotic behaviour depending on how sensitive the system is to the initial conditions.

In the generalized Lagrangian mechanics, the generalized coordinates q (or generalized momenta p) replace the ordinary position (or momentum). Hamiltonian mechanics is slightly different, there are two 1st order equations in the generalized coordinates and momenta:

where q is a tuple of generalized coordinates and similarly p is the tuple of generalized momenta. The initial conditions are similarly defined.

Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a.
From the instantaneous position r = r (t) (instantaneous meaning at an instant value of time t), the instantaneous velocity v = v (t) and acceleration a = a (t) have the general, coordinate-independent definitions;[5]

z The rotational analogues are the angular position (angle the particle rotates about some axis) θ = θ(t), angular velocity ω = ω(t), and angular acceleration a = a(t):

where

is a unit axial vector, pointing parallel to the axis of rotation, = unit vector in direction of r, = unit vector tangential to the angle.
NB: In these rotational definitions, the angle can be any angle about the specified axis of rotation. It is customary to use θ, but this does not have to be the polar angle used in polar coordinate systems.
For a rotating rigid body, the following relations useful for describing the motion hold:

These equations apply to a particle moving linearly, in three dimensions in a straight line, with constant acceleration.[6] Since the vectors are collinear (parallel, and lie on the same line) - only the magnitudes of the vectors are necessary, hence non-bold letters are used for magnitudes, and because the motion is along a straight line, the problem effectively reduces from three dimensions to one.

where r0 and v0 are the particle's initial position and velocity, r, v, a are the final position (displacement), velocity and acceleration of the particle after the time interval.

Here a is constant acceleration, or in the case of bodies moving under the influence of gravity, the standard gravity g is used. Note that each of the equations contains four of the five variables, so in this situation it is sufficient to know three out of the five variables to calculate the remaining two.
SUVAT equations

In elementary physics the above formulae are frequently written:

where u has replaced v0, s replaces r, and s0 = 0. They are often referred to as the "SUVAT" equations, eponymous from to the variables: s = displacement (s0 = initial displacement), u = initial velocity, v = final velocity, a = acceleration, t = time.

Elementary and frequent examples in kinematics involve projectiles, for example a ball thrown upwards into the air. Given initial speed u, one can calculate how high the ball will travel before it begins to fall. The acceleration is local acceleration of gravity g. At this point one must remember that while these quantities appear to be scalars, the direction of displacement, speed and acceleration is important. They could in fact be considered as uni-directional vectors. Choosing s to measure up from the ground, the acceleration a must be in fact −g, since the force of gravity acts downwards and therefore also the acceleration on the ball due to it.

At the highest point, the ball will be at rest: therefore v = 0. Using equation [4] in the set above, we have:

Substituting and cancelling minus signs gives:

Constant circular acceleration
The analogues of the above equations can be written for rotation. Again these axial vectors must all be parallel (to the axis of rotation), so only the magnitudes of the vectors are necessary:

where α is the constant angular acceleration, ω is the angular velocity, ω0 is the initial angular velocity, θ is the angle turned through (angular displacement), θ0 is the initial angle, and t is the time taken to rotate from the initial state to the final state.

General planar motion

These are the kinematic equations for a particle traversing a path in a plane, described by position r = r(t).[9] They are actually no more than the time derivatives of the position vector in plane polar coordinates in the context of physical quantities (like angular velocity ω).

The position, velocity and acceleration of the particle are respectively: where are the polar unit vectors. Notice for a the components (?rω2) and 2ωdr/dt are the centripetal and Coriolis accelerations respectively.

Special cases of motion described be these equations are summarized qualitatively in the table below. Two have already been discussed above, in the cases that either the radial components or the angular components are zero, and the non-zero component of motion describes uniform acceleration.

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