The ancient Greek understanding of physics was limited to the statics of simple machines (the balance of forces), and did not include dynamics or the concept of work.

[5] In 1637, the French philosopher René Descartes wrote:[6] Lifting 100 lb one foot twice over is the same as lifting 200 lb one foot, or 100 lb two feet.In 1686, the German philosopher Gottfried Leibniz wrote:[7] The same force ["work" in modern terms] is necessary to raise body A of 1 pound (libra) to a height of 4 yards (ulnae), as is necessary to raise body B of 4 pounds to a height of 1 yard.In 1759, John Smeaton described a quantity that he called "power" "to signify the exertion of strength, gravitation, impulse, or pressure, as to produce motion."

[9][10][11] Both scientists were pursuing a view of mechanics suitable for studying the dynamics and power of machines, for example steam engines lifting buckets of water out of flooded ore mines.

According to Rene Dugas, French engineer and historian, it is to Solomon of Caux "that we owe the term work in the sense that it is used in mechanics now".

This is approximately the work done lifting a 1 kg object from ground level to over a person's head against the force of gravity.

Conversely, a decrease in kinetic energy is caused by an equal amount of negative work done by the resultant force.

[18] From Newton's second law, it can be shown that work on a free (no fields), rigid (no internal degrees of freedom) body, is equal to the change in kinetic energy Ek corresponding to the linear velocity and angular velocity of that body,

[21] Examples of workless constraints are: rigid interconnections between particles, sliding motion on a frictionless surface, and rolling contact without slipping.

For moving objects, the quantity of work/time (power) is integrated along the trajectory of the point of application of the force.

The small amount of work δW that occurs over an instant of time dt is calculated as

Calculating the work as "force times straight path segment" would only apply in the most simple of circumstances, as noted above.

This integral is computed along the trajectory of the rigid body with an angular velocity ω that varies with time, and is therefore said to be path dependent.

This result can be understood more simply by considering the torque as arising from a force of constant magnitude F, being applied perpendicularly to a lever arm at a distance

Notice that only the component of torque in the direction of the angular velocity vector contributes to the work.

The scalar product of a force F and the velocity v of its point of application defines the power input to a system at an instant of time.

Integration of this power over the trajectory of the point of application, C = x(t), defines the work input to the system by the force.

It is tradition to define this function with a negative sign so that positive work is a reduction in the potential, that is

In the absence of other forces, gravity results in a constant downward acceleration of every freely moving object.

Near Earth's surface the acceleration due to gravity is g = 9.8 m⋅s−2 and the gravitational force on an object of mass m is Fg = mg.

Consider a spring that exerts a horizontal force F = (−kx, 0, 0) that is proportional to its deflection in the x direction independent of how a body moves.

The work of this spring on a body moving along the space with the curve X(t) = (x(t), y(t), z(t)), is calculated using its velocity, v = (vx, vy, vz), to obtain

For convenience, consider contact with the spring occurs at t = 0, then the integral of the product of the distance x and the x-velocity, xvxdt, over time t is ⁠1/2⁠x2.

The derivation of the work–energy principle begins with Newton's second law of motion and the resultant force on a particle.

Computation of the scalar product of the force with the velocity of the particle evaluates the instantaneous power added to the system.

In particle dynamics, a formula equating work applied to a system to its change in kinetic energy is obtained as a first integral of Newton's second law of motion.

Isolate the particle from its environment to expose constraint forces R, then Newton's Law takes the form

Consider the case of a vehicle that starts at rest and coasts down an inclined surface (such as mountain road), the work–energy principle helps compute the minimum distance that the vehicle travels to reach a velocity V, of say 60 mph (88 fps).

Rolling resistance and air drag will slow the vehicle down so the actual distance will be greater than if these forces are neglected.

where V is the magnitude of V. The constraint forces between the vehicle and the road cancel from this equation because R ⋅ V = 0, which means they do no work.

where F and T are the resultant force and torque applied at the reference point d of the moving frame M in the rigid body.