Motion In A Straight Line

Motion In A Straight Line

Distance And Displacement
Many objects that physicists study from atoms to galaxies are in motion. Motion may be orderly or random, steady or intermittent or even a confu- sing mixture of these. We can not hope to gain insight into how nature works unless we can clearly define and then measure motion.
       The subject of Kinematics is concerned with the description of how body moves in space and time. In translational motion all parts of a body undergo the same change in position.
    This chapter is restricted to translational motion along a straight line that is one dimensional motion also known as rectilinear motion.
The objects in motion shall be treated as point objects and this approximation is valid so far as the size of the object is very much smaller compared to the distance travelled by it in a reasonable duration of time.
    The study of motion of bodies without considering the cause of motion is called Kinematics. In the universe every object moves, though some objects appear to be stationary. The terms motion and rest are relative. We begin this chapter with motion along a straight line, i.e. motion in one dimension. Later, we look into motion in a plane, i.e. two-dimensional motion.
REST, MOTION
If a particle's position does not change either with respect to a fixed point or with respect to time, then it is said to be at rest.
    If a particle's position is continuously changing with respect to a fixed point and with respect to able time, then it is said to be in motion.
POSITION, PATH LENGTH
The coordinates of a coordinate system describes the position of an object with respect to the origin of the system.
"Length of actual path between initial and final positions is called distance or path length".
DISPLACEMENT
"The shortest straight line distance directed from initial position to final position irrespective of the path is called displacement". (or) displacement is the change in position.
Let \( x_1 \) and \( x_2\)  be the positions of an object at the instants t, and t, respectively then displacement is\( \Delta x \) =x₂-x₁ in the time interval \( \Delta t\)=t₂-t₁ .
    The 'distance or path length' and 'displacement' are two different quantities. Distance has just a magnitude (numerical value) and no direction. whereas, the displacement has magnitude as well as direction and obeys the laws of vector addition. Such quantities are represented as vectors. Here distance is scalar and displacement is vector.

For example, displacement of the car in moving from O to P is:

        \(\Delta{x}\)=x₂- x₁= (+360m)-0m =+360m
The displacement has a magnitude of 360 m and is directed in the positive x direction as indicated by the + sign.
    Similarly, the displacement of the car from P to Q is  240 m-360 m= 120 m. The negative sign indicates the direction of displacement.
    We should remember The magnitude of displacement may or may not be equal to the path length traversed by an object.
For example, for motion of the car from O to P, the path length is +360 m and the displacement is +360 m. In this case, the magnitude of displacement (360 m) is equal to the path length (360 m).
    But consider the motion of the car from 0 to P and back to Q. In this case, the path length (+360 m)+(+120 m)=+ 480 m. However,the displacement = (+240 m)-(0 m) = + 240 m. Thus, the magnitude of displacement (240 m) is not equal to the path length (480 m).
Application-1:

Consider a particle moving from A to B along a curve as shown.
The distance travelled is equal to the length of the curve AB, whereas the magnitude of the displacement is equal to the length of the straight
Application -2:
            
If a person walks from A to B and then from B to C as shown,
Path length travelled    AB+ BC= 7m
Displacement        AC=5m
Application-3:
A particle moves over an arc PQ, of a circle of radius R, subtending an angle at the centre.

a) distance travelled = arc PQ = R\(\theta\)
b) displacement=straight line PQ = 2R sin\(\theta \)/2