# 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

1
by 2092000

2015-04-02T01:09:59+05:30

### This Is a Certified Answer

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
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.
====================================
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.