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Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456 Chapter Contents Position, Path length and Displacement Average velocity and Average speed Instantaneous velocity and speed Acceleration Kinematic equations of uniformly accelerated motion Graphs Motion under gravity Relative velocity Chapter 3 Motion of a body in a straight line is also termed as rectilinear motion. Only one position co-ordinate is required to describe the one dimensional motion. POSITION, PATH LENGTH AND DISPLACEMENT (i) Displacement : The shortest distance between the initial and final positions is known as the magnitude of displacement and it is directed from initial to final position. Following points should be noted about displacement. (a) It may be positive, negative or zero. (b) It is the vector from initial position to the final position of the object. (c) It is not affected by the shift of position of the origin of the coordinate axes. (d) Displacement of an object is independent of the path followed by the object. (e) It has units of length. (ii) Distance : The total length of actual path traversed by the body between initial and final positions is called distance. (a) It is a scalar quantity. (b) It may be positive or zero (if object at rest), but never be negative. (c) Distance  (displacement) (d) In straight line motion in one direction distance and magnitude of displacement are equal. (e) Distance travelled between two positions gives length of actual path while displacement is length of unique path. (f) For a moving particle, distance never decreases with time. Motion in a Straight Line
38 Motion in a Straight Line NEET Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph.011-47623456 Example 1 : Displacement of a person moving from A to B along a semicircular path of radius R is 100 m. What is the distance travelled by him? Solution : Displacement = 2R  100 = 2R R  R = 50 A B O The distance travelled by the person is R. So, the distance travelled by the person is 3.14  50 = 157 m AVERAGE VELOCITY AND AVERAGE SPEED Average velocity is defined as the change in the position or displacement of object divided by the time interval in which the displacement occurs. 2 1 2 1 change in position total time taken x x v t t     x1 x2 t1 t2 x t P1 P2 Average speed is defined as the total path length travelled divided by the total time interval during which the motion has taken place. Total path length Average speed = Total time interval Note : (i) Average speed is a scalar while average velocity is a vector both having same units and dimensions. (ii) Average speed of a particle in a given time interval is never less than the magnitude of the average velocity because distance follows scalar addition while displacement follows vector addition. (iii) The magnitude of average velocity in an interval need not be equal to its average speed in that interval. (iv) An object may have varying velocity, without having varying speed, as in case of a uniform circular motion because velocity can change even by changing direction. (v) If velocity is constant, then speed will also be constant, but if the speed is constant then velocity may or may not be constant, as in case of uniform circular motion. (vi) It is not possible to have a situation in which the speed of the particle is never zero but the average speed in an interval is zero. (vii) It is not possible to have a situation in which the speed of a particle is always zero but the average speed is not zero. (viii) Average speed or velocity depends on time interval over which it is defined. (ix) For a given time interval average velocity has single value while average speed can have many values depending on path followed. (x) If after motion the body comes back to its initial position then average velocity is zero but the average speed is greater than zero and finite. (xi) For a moving body average speed can never be negative or zero (unless t   ) while average velocity can be zero. (xii) In general, average speed, is not equal to magnitude of average velocity. However, it can be so if the motion is along a straight line without change in direction.
NEET Motion in a Straight Line 39 Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph.011-47623456 INSTANTANEOUS VELOCITY AND SPEED The instantaneous velocity of an object at a given instant of time is defined as the limit of average velocity as the time interval t, becomes infinitesimally small i.e., 0 lim t x v   t    and the limit of the ratio x/t as t approaches to zero is called the derivative of x w.r.t. t and is written as dx dt . Instantaneous speed is magnitude of instantaneous velocity. Differentiation: Graphically, differentiation means slope of graph at a point on graph. If y = f(x) Then dy dx = differentiation of y w.r.t. x dy dx represents that how fast y change when we change x ( ) 1 n d x n nx dx   ; where n is constant d au du ( ) a dx dx  ; du du dx dt dx dt   d uv dv du ( ) u v dx dx dx   ; 2 d u v du dv (/) 1 v u dx dx dx v         / / du du dx dv dv dx  (sin ) cos d x x dx  ; (cos ) sin d x x dx   2 (tan ) sec d x x dx  ; 2 (cot ) cosec d x x dx   (sec ) tan sec d x x x dx  ; (cosec ) cot cosec d x x x dx   1 ( ) d d n n u u nu dx dx   ; 1 (In ) d u du u  ( ) d u u e e du  Maxima and Minima It is one of the application of differential calculus. Consider a physical quantity y depends on another quantity x as shown below. It is clear from the graph at x = x1 and x = x2 , the value of y is maximum and minimum respectively. Now, at both points A and B the angle of tangent with x-axis is zero. Therefore at both points A and B slope of curve is zero. A B x1 x2 y x O As we know slope of curve gives rate of change of y w.r.t. x. Therefore at x = x1 and x = x2 , the slope of graph is zero.
40 Motion in a Straight Line NEET Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph.011-47623456 Remember : At maxima At minima  0 dy dx  0 dy dx 2 2  0 d y dx 2 2  0 d y dx [where 2 2 ;        d y d dy dx dx dx called second order derivative of y w.r.t. x] Example 2 : The displacement x of an object is given as a function of time, x = 2t + 3t2 Calculate the instantaneous velocity of the object at t = 2 s Solution : x = 2t + 3t 2 2 6 dx v t dt   Substituting the value of t as 2, we get v = 2 + 6  2 = 14 m/s So the instantaneous velocity of the object at time t = 2 s is 14 m/s ACCELERATION Time rate of change of velocity is known as acceleration. It is a vector quantity. If v is the change in velocity in time t, then Average acceleration,    v a t Instantaneous acceleration, 2 2 . dv d x a dt dt   Uniform Acceleration : If the velocity of the body changes in equal amount during each equal time interval; then the acceleration of the body is said to be uniform. Acceleration is uniform when neither its direction nor magnitude change with respect to time. Variable Acceleration : If the velocity of body changes by different amounts during equal time interval, then the acceleration of the body is known as variable acceleration. Acceleration is variable if either its direction or magnitude or both change with respect to time. A good example of variable acceleration is the acceleration in uniform circular motion. Note : (i) Acceleration may result due to the change in the direction of velocity without any change in the magnitude of the velocity (i.e., speed) (ii) If the velocity is zero at an instant, the acceleration need not be zero at that instant as in case of motion under gravity at the topmost point. (iii) If a particle has non-zero acceleration its velocity has to vary. (either in magnitude or in direction or in both) (iv) There is no definite relation between the direction of velocity vector and the direction of acceleration vector. So when a particle is in motion, its acceleration may be in any direction. (v) It is possible that an object can be increasing in speed when its acceleration is decreasing as in case of a raindrop.