# What is Rectilinear Motion & the Mathematical Form of the Motion?

__Rectilinear Motion__

The body is considered to be in motion when the location of an item varies with regard to time and its surroundings. Motion may be defined mathematically using displacement, velocity, and acceleration in a certain frame of reference. A particle’s motion may be classed based on its trajectory, with rectilinear motion being the most basic. The vectors of displacement, velocity, and acceleration are limited to one dimension. Uniform motion (zero acceleration), uniformly accelerated motion (non-zero constant acceleration), and motion with non-uniform acceleration are the three forms of rectilinear motion. Free-fall under gravity and simple harmonic motion of a mass linked to a spring were instances of rectilinear motion.

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__Definition__

Rectilinear (or linear) motion is defined as a particle’s movement constrained to a straight line. Only one coordinate may be used to describe such a motion. The particle’s displacement, as well as its derivatives, such as velocity and acceleration, are all one-dimensional vectors. A automobile going along a straight route in free fall under the gravitational field of the Earth can be modelled as rectilinear movements.

__Mathematical Form of the Motion__

A one-dimensional reference frame with an axis (X-axis) and an origin at O (x = 0) is used to qualitatively evaluate rectilinear motion. Placement, length, & displacement: A particle’s position is indeed a vector quantity that is measured from the origin to the particle. The distance among them determines its magnitude. When you set a particle in motion, it follows a route that changes its location over time. Displacement is the vector difference in position after a time period, pointing from the start to the end location. The overall path covered along the trajectory is measured in distance, whereas the shortest path is measured in displacement. The displacement is provided by, if the particle’s location moves from xi to xf in time Δt, x=x_{f}−x_{i}.

Speed as well as Velocity: Speed refers to the rate of change in distance, whereas velocity refers to the rate of change in displacement over time. While speed is a scalar, velocity is a vector with the same direction as displacement. At time t, the instantaneous velocity is given by, v=(d/dt) x. The time rate of change is defined as acceleration when velocity changes with time. It’s a vector, too, a=(d/dt) v=d^{2}/dt^{2})^{ x}. Because all vectors are confined to one dimension, only the magnitudes need to be considered.

Graphical Representation: When the particle’s location is plotted as a function of time, the graph depicts the particle’s travel. Because velocity is the time derivative of location, the slope of this graph determines velocity at any given moment. Because acceleration is the time derivative of velocity, the slope of the velocity against time graph is used to calculate it.

Rectilinear Motion Formulas: Rectilinear motion may be classified into three forms based on distinct acceleration values: uniform rectilinear motion, evenly accelerated motion, as well as motion with non-uniform velocity. This equation connects location and velocity at any given time. Acceleration may be depicted as a straight line parallel to the time axis since it is constant in time. Velocity is likewise linear, but it fluctuates with time, resulting in a nonzero slope straight line. The trajectory is parabolic and the displacement is quadratic in time.

Non-Uniform Acceleration Motion: In these movements, the acceleration varies with time and place. The magnitude of the acceleration is proportional to the location in simple harmonic motion. An SHM’s trajectory is sinusoidal.

Free fall under gravity, for example, can be represented as a rectilinear motion with constant acceleration if the gravitational acceleration of an item owing to the Earth’s gravitational pull is regarded constant across the distance of interest. Any nonconservative force in the issue, such as air resistance or viscosity, is assumed to be non-existent. When an item falls freely from a height of h just above ground, the starting height d(t = 0) is h, and its initial velocity v(t = 0) is zero. g = 9.8 m/s2 is the constant acceleration. Using position and velocity expressions. v(t)=gt is the velocity at any given time t. d(t)=12gt2 is the displacement at any time t. This downward displacement from the starting height causes the object’s height to diminish over time.

Interesting Facts: A rectilinear motion can be approximated by the motion of two particles under the influence of a central force (e.g., electrostatic force). Because of the rotation of the Earth, free-fall under its gravitational field is not a rectilinear motion. Because of the rotation, the Coriolis force bends the free-fall trajectory. The dynamics of linear motion and rotation (on a plane) around an axis are identical.

Related question: A stone projected vertically up with velocity v from the top of a tower reaches the ground with velocity 2v. The height of the tower is?

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