6. Equations of uniformly accelerated motion

  • There are three equations of bodies moving with uniform acceleration which we can use to solve problems of motion

First Equation of motion

  • The first equation of motion is \(v = u + at\) , where v is the final velocity and u is the initial velocity of the body.
  • First equation of motion gives velocity acquired by body at any time \(t\).
  • Now we know that acceleration
    equation of acceleration
    so, \(a = \frac{{v - u}}{t}\)
    and, \(at = v - u\)
    rearranging above equation we get first equation of motion that is
    \(v = u + at\)

Second Equation of motion

  • Second equation of motion is
    \(s = ut + \frac{1}{2}a{t^2}\)
    where \(u\) is initial velocity, \(a\) is uniform acceleration and \(s\) is the distance travelled by body in time \(t\).
  • Second equation of motion gives distance travelled by a moving body in time \(t\).
  • To obtain second equation of motion consider a body with initial velocity \(u\) moving with acceleration a for time \(t\) its final velocity at this time be \(v\). If body covered distance \(s\) in this time \(t\) , then average velocity of the body would be
    average velocity
    Distance travelled by the body is

    From first equation of motion
    \(v = u + at\)
    So putting first equation of motion in above equation we get ,
    \(s = \frac{{u + u + at}}{2} \times t = \frac{{\left( {2u + at} \right)t}}{2} = \frac{{2ut + a{t^2}}}{2}\)
    Rearranging it we get
    \(s = ut + \frac{1}{2}a{t^2}\)

Third equation of motion

  • Third equation of motion is
    \({v^2} = {u^2} + 2as\) where \(u\) is initial velocity, \(v\) is the final velocity, \(a\) is uniform acceleration and \(s\) is the distance travelled by the body.
  • This equation gives the velocity acquired by the body in travelling a distance \(s\).
  • Third equation of motion can be obtained by eliminating time t between first and second equations of motion.
    So, first and second equations of motion respectively are
    \(v = u + at\) and \(s = ut + \frac{1}{2}a{t^2}\)
    Rearranging first equation of motion to find time t we get
    \(t = \frac{{v - u}}{a}\)
    Putting this value of t in second equation of motion we get
    \(s = \frac{{u\left( {v - u} \right)}}{a} + \frac{1}{2}a{\left( {\frac{{v - u}}{a}} \right)^2}\)

    \(s = \frac{{uv - {u^2}}}{a} + \frac{{a\left( {{v^2} + {u^2} - 2uv} \right)}}{{2{a^2}}}\)

    \(s = \frac{{2uv - 2{u^2} + {v^2} + {u^2} - 2uv}}{{2a}}\)

    Rearranging it we get
    \({v^2} = {u^2} + 2as\)
  • These three equations of motion are used to solve uniformly accelerated motion problems and following three important points should be remembered while solving problems
    1. if a body starts moving from rest its initial velocity \(u = 0\)
    2. if a body comes to rest i.e., it stops then its final velocity would be \(v = 0\)
    3. If a body moves with uniform velocity then its acceleration would be zero.

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