Question

a particle moves in a straight line in such a manner that s=1/2vt, s being the distance travelled in time t and v the velocity at the end of time t. prove that the acceleration is constant

Answer 1

motion in a single direction.
     s = v t / 2

 velocity = ds / dt
          v  = 1/2 [ v * 1 +  t  dv/dt ]
         v = 1/2  [ v + a t ]
         1/2 v = 1/2 at
         v = a t

  acceleration a = dv/dt = a * 1 + t * d a / dt
                 a = a + t * da/dt
     => da / dt = 0
     => a = dv/dt = acceleration  = constant.
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we can perhaps do the following way:
   if acceleration is constant then, the equations of motion are:
        v = u + at
         s = u t + 1/2 a t²
           v² - u² = 2 a s

  in our context,  s = 1/2  v t
         v = 2 s / t,    differentiate wrt  time t:
         a = dv/dt
            = 2 [t * ds/dt - s * 1 ] / t²
            = 2 [ v t - s ] / t²
            = 2 [ v t - 1/2 vt ] / t²
             = 2 * vt/2  /t²
             = v / t
         da/dt = [ t * dv/dt - v * 1 ] /t²
                  = [ t * a - v ] / t²
                   = 0          as  v = a t
 Hence, Acceleration is constant.  as its derivative is zero.