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Linear motion

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In physics, and more particularly in mechanics, a linear motion is a movement that takes place along a straight line. During a rectilinear motion, the velocity vector v retains its direction, its value being able to remain constant (uniform linear motion) or to vary (non-uniform linear motion).

In the theoretical case of the motion of a point object, the motion can be fully described by a one-dimensional equation typically x = f(t) where x is the position of the system and t is the time. In the concrete case of a non-punctual object, we often reduce the study of the system to that of its center of inertia, neglecting in this case all the effects of the possible rotations of the object.

Kinematics of the point

Considering a point M in rectilinear motion along an axis (O,i), M is indicated by its abscissa x(t): OM = x(t) i

Displacement of the system: The displacement is the distance traveled at the speed v by the system during a duration Δt. His unit is the meter (m) in the International System of Units. When the system goes from a position M1 to the position M2, its displacement is equal to x2 – x1.

Speed: The speed of a point M is the derivative of the vector position OM with respect to time v = dOM/dt. In the case of a rectilinear movement, OM = x(t) i, it comes:

v(t) = dx(t)/dt i

This implies that the direction of v(t).

Constant speed: If the displacement is at a constant speed, the movement is said to be rectilinear. The speed is then the ratio of the length of any displacement to the duration of this displacement:

v(t) = (x2 – x1)/(t2 – t1)

Acceleration: The acceleration of a point M is the derivative of the velocity vector v with respect to the time a(t) = dv(t)/dt = dv(t)/dt i. If the motion of M is rectilinear then the direction of the vector a(t) remains constant so a = dv(t)/dt.

Forces and rectilinear movement

Uniform rectilinear movement: According to Newton’s first law, if there is no force exerted on a body (isolated body), or if the sum of the forces exerted on it is zero (pseudo-isolated body) , then its movement in a Galilean frame will be at once rectilinear (constant direction of velocity) and uniform (value of constant velocity).

Non-uniform rectilinear motion: According to Newton’s second law, if the mass of the body is constant, the acceleration undergone by this body in a Galilean frame of reference is proportional to the resultant of the forces it undergoes, and inversely proportional to its mass m.

a = 1/m ΣF

Acceleration is the derivative of speed over time:

a = dv/dt


dv/dt = 1/m ΣF

The derivative of the velocity vector is collinear with the resultant forces so if the resultant of the forces remains in the direction of v then the direction of the vector v does not change and the motion is straight.

Propagation of light

The light has a rectilinear (and uniform) motion in any homogeneous transparent medium, especially vacuum or very dry air. This property is at the base of geometrical optics.

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