Motion with constant acceleration

6. Motion with constant acceleration

  • Motion with constant acceleration or uniformly accelerated motion is that in which velocity changes at the same rate throughout motion.
  • When the acceleration of the moving object is constant its average acceleration and instantaneous acceleration are equal. Thus from eq. 5 we have
    Motion with constant acceleration
  • Let v0 be the velocity at initial time t=0 and v be the velocity of object at some other instant of time say at t2=t then above eq. 7 becomes
    uniformly accelerated motion
    or , v=v0+at                         (8)
  • Graphically this relation is represented in figure 8 given below.
    graph between position and  time in rectilinear motion with constant acceleration
  • Thus from the graph it can be seen clearly that velocity v at time t is equal to the velocity v0 at time t=0 plus the change in velocity (at).
  • In the same way average velocity can be written as
    uniformly accelerated motion
    where x0 is the position of object at time t=0 and vavg is the averag velocity between time t=0 to time t.The above equation then gives
    x=x0+vavgt                    (9)
    but for the interval t=0 to t the average velocity is
    uniformly accelerated motion
    Now from eq. 8 we find
    vavg = v0 + ½(at)                    (11)
    putting this in eq. 9 we find
    x = x0 + v0t + ½(at2)
    x - x0 = v0t + ½(at2)                (12)
    this is the position time relation for object having uniformly accelerated motion.
  • From eq. 12 it is clear that an object at any time t has quadratic dependence on time, when it moves with constant acceleration along a straight line and x-t graph for such motion will be parabolic in natures shown below.

    velocity time graph for rectilinear motion with constant acceleration

  • Equation 8 and 12 are basic equations for constant acceleration and these two equations can be combined to get yet another relation for x , v and a eliminating t so, from 8
    Motion with constant acceleration
    putting this value of t in equation 12 and solving it we finally get,
    v2 = (v0)2 + 2a ( x - x0 )                     (13)
  • Thus from equation 13 we see that it is velocity displacement relation between velocities of object moving with constant acceleration at time t and t=0 and their corresponding positions at these intervals of time.
  • This relation 13 is helpful when we do not know time t.
  • Likewise we can also eliminate the acceleration between equation 8 and 12. Thus from equation 8

    Motion with constant acceleration
    putting this value of a in equation 12 and solving it we finally get,
    ( x - x0 ) = ½ ( v0 + v ) t               (14)
  • Same way we can also eliminate v0 using equation 8 and 12. Now from equation 8
    v0 = v - at
    putting this value of v0 in equation 12 and solving it we finally get,
    ( x - x0 ) = vt + ½ ( at2 )                (15)
    thus equation 15 does not involve initial velocity v0
  • Thus these basic equations 8 and 12 , and derived equations 13, 14, and 15 can be used to solve constant acceleration problems.

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