Motion: An
object is said to be motion if it changes its position w.r.t. some stationary
object with the passage of time. Uniform Motion:
An object is said to be in uniform motion if it covers equal distances in
equal intervals of time. Non-uniform Motion: On the other hand, if a body covers unequal distances is equal intervals of time, then it will be said to be in non-uniform motion. Motion in One Dimension: The change in position of an object with the passage of time means change in its x, y & z coordinates. If only any one of the co-ordinates changes, then motion is called as one-dimensional. On the other hand, motion are called two or three-dimensional depending upon change in two or all the three co-ordinates. Motion of an insect crawling over a floor & a kite flying in the sky are the examples of two & three-dimensional motion respectively. If , we consider a body (say a car ) traveling along a straight line with uniform motion, then we need its different position at different time to describe the motion graphically. As only one of its co-ordinate changes, therefore, we need only one axis. The more quantities to be chosen are: (I) Origin for position & direction for position measurement(II) Origin for time & sense of passage of time To explain fully, we take an example of car starting from a
city O. it reaches at city A in 3 hrs after covering 60 Km, then it reaches at B in 1 hrs
after covering 20 Kms & finally it reaches to P in 2 hrs after covering 50
Kms. If we take ‘O’ as the origin If we take ‘A’ as
origin (II) Origin for time & sense of passage of time: If we take the time, when the car starts from ‘O’ as origin: If we take the time, when the car reaches to ‘A’, as origin, then: The information regarding position & time can also be shown over a single axis altogether, in which we can choose same point as the origin for position & time OR we can choose different points also as their origins. Displacement: Suppose a body starts from A, and following Paths AP, PQ & QB , reaches to B, then we can say that its Final position is B & initial position as A. The difference between B A along AB is called as its displacement. Therefore “ The displacement of a particle is defined as the change of the position of a particle in a particular direction”. Displacement is a vector quantity, as it possesses both, the magnitude & the direction. It is shown as AB But if its position from B to A, then displacement will becomeBA = - AB If the body returns to its initial position then displacement is AA = OOn the other hand, if we measure AP + PQ + QB, then it will be the distance covered by the body to reach at B. There fore “The length of the actual path between initial & final position of the particle is called the distance covered by it”. Distance is a scalar quantity Characteristics of Displacement 1. Displacement has unit of length. 2. Displacement is the shortest distance between two points. 3. Displacement can be +ve, -ve or zero. 4. Displacement doesn’t tell anything about actual path followed by body to reach to final point from initial point. 5. Displacement between two points is a unique path. Velocity: “The time rate of changes of displacement of an object is called the velocity of the object.” Its SI unit is ms-1. Uniform Velocity: when the object undergoes equal displacement in equal interval of time, it is said to be having uniform velocity. Variable Velocity: When either the speed of object or its direction of motion or both changes with time then it is said to be having variable velocity. Kinematics of uniform motion: Suppose a body is traveling along a straight fine with uniform velocity V, origin for position is at ‘O’& origin for time is at ‘A’. Suppose OA = xo, OB = x & OC = X, xo is initial distance of object from origin, then after time ‘t’ it reaches at B such that x = xo +AB or x = xo
+vt ----(1) [using distance = speed x time] When it reaches at C after t’ time then `x’ = xo + vt’ -------- (2) Subtracting (1) from (2), we get x’ – x = ( xo +vt’) – (xo + vt)x’ – x = v(t’ – t) ---------- (3) So Equation (1), (2) & (3), all represents kinematics of uniform motion along straight line. Q.1. A body travels from A to B at 40 ms-1 & from B to A at 60 ms-1. Calculate the average speed and average velocity. Q.2. A man walks on a straight road form home to a market 2.5 Km away with speed of 5 kmh-1. Find the market closed he instantly turns & walks back home with a speed of 7.5 kmh -1. What is the magnitude of average velocity and average speed of the man over the interval of time (i) o to 30 min (ii) 0 to 50 min (iii)o to 40 min. Q.3 A woman starts from her home at 9.00 A.M., walks with a speed of 5 kmh-1 on a straight road upto her office 2.5 km away, stays at the office upto 5.00 P.M. & returns home by an auto with a speed of 25 kmh-1. Choose suitable scales & plot the x – t graph of her motion. Velocity – Time graph for uniform motion: “Velocity – Time graph for an object in uniform motion along a straight path is a straight line parallel to time axis.” “The area covered between the velocity time graph & time axis between given time instants gives the Magnitude of displacement.” Position – Time graph for uniform motion: “The Position – Time graph for an object in uniform motion along a straight path is a straight line inclined to the time axis.” “The slope of the position – time
graph gives the velocity of the body.” Relative velocity: If two trains are running on two parallel tracks with same velocity, in the same direction. Then for the observer of any of the train appears to be at rest. Means its velocity appears to be zero while velocities of both the trains w.r.t ground are not zero. The velocity of one train is zero only w.r.t other train moving with same velocity in the same direction. The velocity of one train is called its relative velocity w.r.t. other train. “ The relative velocity of one body w. r. t. another body is defined as the displacement of the one body w, r, t, another body per unit time.” Its S.I. unit is m/s. If two objects P & P’ have some initial distance xo & xo’ respectively have velocities v & v’ respectively then after time ‘t’, their distance will be given by kinematics: x = xo + vt --------------- (1) x’ = xo’ + vt’ -------------- (2) Subtracting (1) from (2), we get x’ – x = (xo’ - xo) + (v’ – v) t this is the Kinematics of relative motion of p’ w.r.t. P Here x’- x is the relative displacement of object P from P’ and v’ – v is relative velocity of object P w.r.t. P’ (x’ – x) & (v’ – v) may positive, zero or negative. (i) If v’ – v is positive then x’ – x will be positive, & will keep on increasing by the passage of time (ii) If v’ – v is zero, then x’ – x = xo’ – x. That mean’s relative displacement will remain constant. (iii) If v’ – v is negative, then x’ – x will go on decreasing. (iv) If x’ – x is zero, this means that both the objects are running side-by-side. Position – Time graph for relative velocity: If two bodies are running with same velocities then their position – time graph will be parallel straight lines while in case of different velocities, the two position time graphs will cut each at some point. The point of intersection of these two graphs gives the time when the two bodies overtake & at what distance after they start. Case I : When VA and VB are equal. Relative velocity of A w.r.t. B is equal to zero. Case II : When VA and VB are unequal. VA >VB Relative velocity of A w.r.t. B is positive. Case III : When VA and VB are unequal. VA <VB Relative velocity of A w.r.t. B is negative. |