Linear motion

Linear motion, also called rectilinear motion, is one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion with constant velocity or zero acceleration; and non-uniform linear motion with variable velocity or non-zero acceleration. The motion of a particle (a point-like object) along a line can be described by its position formula_1, which varies with formula_2 (time). An example of linear motion is an athlete running 100m along a straight track.
Linear motion is the most basic of all motion. According to Newton's first law of motion, objects that do not experience any net force will continue to move in a straight line with a constant velocity until they are subjected to a net force. Under everyday circumstances, external forces such as gravity and friction can cause an object to change the direction of its motion, so that its motion cannot be described as linear.
One may compare linear motion to general motion. In general motion, a particle's position and velocity are described by vectors, which have a magnitude and direction. In linear motion, the directions of all the vectors describing the system are equal and constant which means the objects move along the same axis and do not change direction. The analysis of such systems may therefore be simplified by neglecting the direction components of the vectors involved and dealing only with the magnitude.
Displacement.
The motion in which all the particles of a body move through the same distance in the same time is called translatory motion. There are two types of translatory motions: rectilinear motion; curvilinear motion. Since linear motion is a motion in a single dimension, the distance traveled by an object in particular direction is the same as displacement. The SI unit of displacement is the metre. If formula_3 is the initial position of an object and formula_4 is the final position, then mathematically the displacement is given by:
formula_5
The equivalent of displacement in rotational motion is the angular displacement formula_6 measured in radians.
The displacement of an object cannot be greater than the distance because it is also a distance but the shortest one. Consider a person travelling to work daily. Overall displacement when he returns home is zero, since the person ends up back where he started, but the distance travelled is clearly not zero.
Velocity.
Velocity refers to a displacement in one direction with respect to an interval of time. It is defined as the rate of change of displacement over change in time. Velocity is a vectorial quantity, representing a direction and a magnitude of movement. The magnitude of a velocity is called speed. The SI unit of speed is formula_7 that is metre per second.
Average velocity.
The average velocity of a moving body is its total displacement divided by total time needed to reach a body from initial point to final point. It is an estimate velocity for a distance to travel. Mathematically, it is given by:
where:
The magnitude of the average velocity formula_13 is called an average speed.
Instantaneous velocity.
In contrast to an average velocity, referring to the overall motion in a finite time interval, the instantaneous velocity of an object describes the state of motion at a specific point in time. It is defined by letting the length of the time interval formula_14 tend to zero, that is, the velocity is the time derivative of the displacement as a function of time.
formula_15 formula_16
The magnitude of the instantaneous velocity formula_17 is called the instantaneous speed.
Acceleration.
Acceleration is defined as the rate of change of velocity with respect to time. Acceleration is the second derivative of displacement i.e. acceleration can be found by differentiating position with respect to time twice or differentiating velocity with respect to time once. The SI unit of acceleration is formula_18 or metre per second squared.
If formula_19 is the average acceleration and formula_20 is the change in velocity over the time interval formula_14 then mathematically,
formula_22
The instantaneous acceleration is the limit, as formula_14 approaches zero, of the ratio formula_24 and formula_14, i.e.,
formula_26
Jerk.
The rate of change of acceleration, the third derivative of displacement is known as jerk. The SI unit of jerk is formula_27. In the UK jerk is also known as jolt.
Jounce.
The rate of change of jerk, the fourth derivative of displacement is known as jounce. The SI unit of jounce is formula_28 which can be pronounced as "metres per quartic second".
Equations of kinematics.
In case of constant acceleration, the four physical quantities acceleration, velocity, time and displacement can be related by using the Equations of motion
here,<br>
formula_33 is the initial velocity<br>
formula_34 is the final velocity<br>
formula_35 is the acceleration<br>
formula_36 is the displacement<br>
formula_37 is the time
These relationships can be demonstrated graphically. The gradient of a line on a displacement time graph represents the velocity. The gradient of the velocity time graph gives the acceleration while the area under the velocity time graph gives the displacement. The area under an acceleration time graph gives the change in velocity.
Analogy with rotational motion.
The following table refers to rotation of a rigid body about a fixed axis: formula_38 is arclength, formula_39 is the distance from the axis to any point, and formula_40 is the tangential acceleration, which is the component of the acceleration that is "parallel" to the motion. In contrast, the centripetal acceleration, formula_41, is "perpendicular" to the motion. The component of the force parallel to the motion, or equivalently, "perpendicular" to the line connecting the point of application to the axis is formula_42. The sum is over formula_43 particles and/or points of application.
The following table shows the analogy in derived SI units: