Mechanics - basic concepts

Mechanics begins with two quantities that describe how an object moves.

Velocity is the change in displacement per unit of time. It is measured in metres per second.

v = Δs / Δt

Here v is velocity in m·s⁻¹, s is displacement in metres, and t is time in seconds.

Acceleration is the change of velocity per unit of time. It is measured in metres per second squared.

a = Δv / Δt

Here a is acceleration in m·s⁻², and Δv is the change in velocity in m·s⁻¹.

Force is any interaction that, when unopposed, will change the motion of an object. Force is measured in newtons.

Momentum is the property of an object with mass that is in motion. It is the product of the mass and the velocity of the object.

p = m · v

Momentum p is measured in kg·m·s⁻¹, which is the same unit as the newton second (N·s). Mass m is in kilograms and velocity v in m·s⁻¹.

Impulse is the change in momentum of an object. An impulse is what an object receives when a force acts on it for a period of time.

Impulse = F · Δt = m · Δv

Impulse is measured in newton seconds (N·s), the same unit as momentum, because impulse and momentum are the same physical quantity seen from two sides.

Uniform accelerated motion is motion in which the velocity of an object changes by an equal amount in every equal period of time. Equal distances are covered in equal intervals of time.

Non-uniform accelerated motion is motion in which unequal distances are covered in equal intervals of time.

Newton's first law. In an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless it is acted upon by a force. In other words, velocity stays constant when no outside forces act.

Newton's second law. In an inertial reference frame, the vector sum of the external forces F on an object is equal to the mass m of that object multiplied by the acceleration a of the object.

F = m · a

The second law can also be written in terms of momentum, and this is the more general form:

F = Δp / Δt

The resulting force is equal to the rate of change of momentum.

Newton's third law. When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body. For every action there is an equal and opposite reaction, written as action = reactio.

F₁ = -F₂

Work is a measure of energy transfer that occurs when an object is moved over a distance by an external force applied in the direction of the displacement. Work is done by a force on a mass when the force and the displacement are parallel.

W = F · d

Work is measured in joules, which is the same as the newton metre (N·m). Here d is displacement in metres.

Energy is the quantitative property that must be transferred to an object in order to perform work on it, or to heat it. The amount of energy transferred to an object equals the work done on the object. Energy is measured in joules (J).

Forms of energy include kinetic, thermal, sound, light, chemical and elastic potential energy.

Power is the amount of energy transferred per unit of time, that is, the rate of energy transfer.

P = W / Δt

Power P is measured in watts (W), work W in joules, and time t in seconds.

The law of conservation of energy states that the total energy of an isolated system remains constant. Energy cannot be created or destroyed, only changed from one form into another.

This lets the two mechanical energies be equated to each other. Kinetic energy and potential energy are:

Eₖ = ½ m v²
Eₚ = m · g · h

So for a body falling freely, ½ m v² = m · g · h. Here Eₖ is kinetic energy, Eₚ is potential energy, g is the acceleration due to gravity and h is the height.

The law of conservation of momentum states that the total momentum of an isolated system remains constant. If two objects collide, the total momentum before the collision is the same as the total momentum after the collision, provided no external forces act on the system.

momentum before = momentum after
m₁ · Δv₁ = m₂ · Δv₂