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1.2 Forces; Newton's second law

The quantity mv is called the momentum. Typically, the mass m is constant in time, and Newton's law can be written in the simplified form

where a is the acceleration, as defined above. It is not always the case that m is independent of t. For example, the mass of a rocket decreases as its propellant is ejected. Under such circumstances, the above equation is incorrect and the full form of Newton's second law must be used.

Newton's second law is insufficient to describe the motion of a particle. In addition, it requires a value for F, obtained by considering the particular physical entities with which the particle is interacting. For example, a typical resistive force may be modelled as a function of the velocity of the particle, for example:

with λ a positive constant. Once independent relations for each force acting on a particle are available, they can be substituted into Newton's second law to obtain an ordinary differential equation, which is called the equation of motion. Continuing the example, assume that friction is the only force acting on the particle. Then the equation of motion is

where v₀ is the initial velocity. This means that the velocity of this particle decays exponentially to zero as time progresses. This expression can be further integrated to obtain the position r of the particle as a function of time.

Important forces include the gravitational force and the Lorentz force for electromagnetism. In addition, Newton's third law can sometimes be used to deduce the forces acting on a particle: if it is known that particle A exerts a force F on another particle B, it follows that B must exert an equal and opposite reaction force, -F, on A.

If a force F is applied to a particle that achieves a displacement δr, the work done by the force is the scalar quantity

Two reference frames S and S, with S traveling at a relative speed of u to S; an event has space-time coordinates of (x,y,z,t) in S and (x,y,z,t) in S.

The space-time coordinates of an event in Galilean-Newtonian relativity are governed by the set of formulas which defines a group transformation known as the Galilean transformation:

Assuming time is considered an absolute in all reference frames, the relationship between space-time coordinates in reference frames differing by a relative speed of u in the x direction (let x = ut when x = 0) is:

x = x - ut

y = y

z = z

t = t

The set of formulas defines a group transformation known as the Galilean transformation (informally, the Galilean transform).

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Topics: Classical Mechanics Particle Objects Force Velocity Law Time...

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