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Rectilinear Motion With, , Variable Acceleration, , , , 11 Introduction, , , , Ww a point (or particle) moves along a straight line, its motion is said to be a, rectilinear motion. . (Meerut 2004, 07, 08), , Here in this chapter we shall discuss the motion of a point (or particle) along a straight, line which may be either horizontal or vertical., , 12 Velocity and Acceleration, , , , Suppose a particle moves along a straight line OX where O rexl, is a fixed point on the line. Let P be the position of the @ x x, Particle at time ¢, where OP = x. If r denotes the position, , vector of P and i denotes the unit vector along OX, then r = OP = xi., , Let v be the velocity vector of the particle at P. Then, , , , , , re, , Scanned with CamScanner
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Kiskea's T.B. Mechanics (Unified), , CT LL AT TT LETTE,, , , , , , , , dr_d dx di_ de, =f 22 y= 5 fuci, ve at a ) at ae at’, , because i is a constant vector. Obviously, the vector v is collinear with the vector i. Thus, for a particle moving along a straight line the direction of velocity is always along the, line itself. If at P the particle be moving in the direction of x increasing (ie.,in the, direction OX ) and if the magnitude of its velocity i.e., its speed be v, we have, , v=vi= de i. Therefore a vy., dt, , dt, On the other hand if at P the particle be moving in the direction of x decreasing (i.¢.,in, the direction XO) and if the magnitude of its velocity be v, we have, , v=-vi= 2 i. Therefore, cu =-v., , dt, Remember: In the case ofa rectilinear motion the velocity of a particle at time t is dx / dt along, the line itself and is taken with positive or negative sign according as the particle is moving in the, direction of x increasing or x decreasing. :, Now let a be the acceleration vector of the particle at P. Then, dy_d(de.)_dx,, , ae a, Thus the vectora is collinear with i i.e.,the direction of acceleration is always along the, line itself. If at P the acceleration be acting in the direction of x increasing and if its, , Di 2, magnitude be f,we have a = f i= S i. Therefore = = f.On the other hand if at P, ft t, , the acceleration be acting in the direction of x decreasing and if its magnitude be f,we, , have, 2, , 2, a=- fi= gh therefore oooh, , Remember: In the case of a rectilinear motion the acceleration of a particle at time tis d?x/at?, , along the line itself and is taken with positive or negative sign according as it acts in the direction ofx, increasing or x decreasing., , Since the acceleration is produced by the force, therefore while considering the sign of, d?.x/dt? we must notice the direction of the acting force and not the direction inwhich, , the particle is moving. For example if the direction of the acting force is that of ¥, i ‘ 2 2 waeee, increasing, then d*x/dt“ must be taken with positive sign whether the particle is, , moving in the direction of x increasing or in the direction of x decreasing., Other Expressions for acceleration :, , et — y= a - We can then write, a, , dx a (de) ae ak dy, , gl a a eae ae, , , , Scanned with CamScanner
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Rectllinear Motion with Varlable Acceleration, , Sa, , , , “OD, d@x dv, , dv, Thus Fa and p me three expressions for representing the acceleration and any, , one of them may be used to suit the convenience in working out the problems., , Inustrative a, , Exampie 1: Ifat time t the displacement x of a particle moving awa ay from the origin is given by, x=asint +b cost, find the velocity and acceleration of the particle., , Solution: Given that x=asint +b cost., Differentiating w.r.t. ‘t’, we get, , . dx, the velocity v = ae O08 t—bsint., , Differentiating again, we have, , : dv :, the acceleration = a asint—bcost=-x. |, iE, , A point moves in a straight line so that its distance ‘sfrom «fixed point atany timet, is ieproporiiondl tot”. If v be the velocity and f the acceleration at any time t, show that, , LP = nfs/(n-)). Cae, Solution: Here, distance s <t”. Wz, , let s=kt", (1), where k is a constant of proportionality., , Differentiating (1), w.r.t. ‘t’, we have, the velocity v = ds/dt = int", exit2), , , , Again differentiating (2),, the acceleration f = ° = kn (n-1) "7 . (3), P =(knt" PY =e en?, , _art{kn(n-)) t"?} ke", “al ‘, (1-1) &, , , , = afs , substituting from (1) and (3)., (n-1), , Example 3; The law of motion in a straight line being given by s = vt, prove that the, , acceleration is constant., , Solution! We haves=4y=1 4), “ops +, : 2 2 dt ., , , , Scanned with CamScanner
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Aloe) Kiskea's T.B. Mechanics (Unified), , Differentiating w.r.t. ‘t’, we get, , 2, ds_ 1 ds 1 ds lds _1 ds ds_ d's,, Sof 8s eg sor SSS at oF q, dt 2d? 2de 2dt 2dt dt dt, Differentiating again w.r.t. t, we get, : 3, @s_ ds ds, or #8 429 or #8 20, because t #0., dt? dt? de? dt? dt, 3 2 2, Now as L o> a as =0 > es = constant., at3 dt \de* dt, , Hence the acceleration is constant., ~ Example 4: Prove that if a point moves with a velocity varying as any power (not less than, unity) of its distance from afixed point which itis approaching, it will never reach that point., , Solution: If.xis the distance of the particle from the fixed point Oat any tim, " wherek isaconstant and mis not less than 1., , et, thenits, , speed vat this time is given by» =k x, , Since the particle is moving towards the fixed point i¢.,in the direction x decreasing,, therefore, , dx/dt=-v, or dx/dt =- ke". ~(l), Case I. Ifn=),then from (1), we have, , dx/dt =—ke, , 1 de, , dt=-=—, or kx, , Integrating, ¢ = - (I/k ) log x+ A, where A is a constant., , Putting x =0, the time ¢ to reach the fixed point O is given by, t=-(I/k)log0 + A=, , ie., the particle will never reach the fixed point O., , Case II. Ifm>1, then from (1), we have, , , , 1, dt=-—x" dx., k, -n+l, Integrating, ¢=—- Fa A + B, where B is a constant, , 1, , ¢=—_—_—__=>_.., k(n-l)x" PB:, , or, , Putting x = 0, the time ¢ to reach the fixed point O is given by, toot B=eo :, , ie., the particle will never reach the fixed point O., Hence if 12 |, the particle will never reach the fixed point, it is approaching, ritisa aching,, , Example 5: Uf t be reganed as « sfnsctlon of velocity v, prove that the rat of decrease of, J eof alec, , acceleration is given by f Std”), f being the acceleration, h (Kanpur 2007), , Scanned with CamScanner
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\\ to EE Wea _, , Rectilinear Motion with Varlable Acceleration, , , , , , , , Solution: Let f be the acceleration at time t. Then f = dv/dt -Now the rate of decrease, of acceleration = — df /dt, , -4(B)--4 (4) regarding ¢ as a function of, dt dt) de ay) * TBAB £48 2 tunction oly, , - 4 (ay (ay dt dv, dv \dv dt \dv}) dv? dt, (tf ae Be (a Be pa,, dt} dt dv? \dt) ay? dy?, , , , , , , , , , ee (‘Comprehensive Exercise 1, , WW Aparticle moves alonga straight line such that its displacement x, froma point on, the line at time t, is given by, , x=t3 907 424t4+6., , Determine (i) the instant when the acceleration becomes zero, (ii) the position of, the particle at that instant and (iii) the velocity of the particle, then., , 2, Aparticle moves along a straight line and its distance from a fixed point on the, line is given by x = a cos (it + £). Show that its acceleration varies as the distance, from the origin and is directed towards the origin., , 3~ A particle moves alonga straight line such that its distance x from a fixed point on, it and the velocity v there are related by v =p (a2 -2). Prove that the, , acceleration varies as the distance of the particle from the origin and is directed, towards the origin., , 4 The velocity of a particle moving along a straight line, when at a distance x from, the origin (centre of force) varies as Vv (a? ae y/ xe }.Find the law of acceleration., , 5.A point moves ina straight line so that its distance froma fixed point in that line is, the square root of the quadratic function of the time ; prove that its acceleration, varies inversely as the cube of the distance from the fixed point., , A If a point moves in a straight line in such a manner that its retardation is, proportional to its speed, prove that the space described in any time is, proportional to the speed destroyed in that time., , XW The velocity of a particle moving along a straight line is given by the relation, =a x +2 bx +c Prove that the acceleration varies as the distance froma fixed, , point in the line., , ° (Answers J, , , , , , 1. (i) 3 seconds (ii) 24 units (iii) 3 units in the, , 4. Accel, varies inversely as the cube of the distance, towards the origin,, , direction of wv increasing., , from the origin and is directed, , PE, , Scanned with CamScanner
