# Starting from rest, a body moves at first witha constant accelaration a. Then it moves with a uniform velocity and finally with a constant retardation a before coming to rest. If the displacement is s, and time taen is t, prove that the body was in motion with constant velocity for a time interval of root[(t2-4s)/a].

Dear student

In the question particle has three motion
1) Accelerated motion
2) Uniform motion
3) Decelerated motion

1) Accelerated motion : Let s1 be the distance covered by the particle in t1 time-interval. Since particle starts from rest, u=0 and a is the acceleration so from equation of motion
$s=ut+\frac{1}{2}a{t}^{2}$
.................(1)
and the velocity of the particle at the end of the time interval is given by

$v=u+at$
$⇒v=a{t}_{1}$                      ..................(2)

2) Uniform motion : Let ​s2 be the distance covered by the particle in ttime-interval with uniform velocity v so

${s}_{2}=v{t}_{2}$
$⇒{s}_{2}=a{t}_{1}{t}_{2}$                 ...................(3)

3) Decelerated motion : ​Let ​s3 be the distance covered by the particle in t3 time-interval with deceleration a. The velocity of the particle at the start of time interval ​tis v and final velocity is 0 so

$⇒{t}_{1}={t}_{3}$                  ..................(4)
and ​s3 is given by
${s}_{3}=v{t}_{3}-\frac{1}{2}a{t}_{3}^{2}$
${s}_{3}=a{t}_{1}{t}_{3}-\frac{1}{2}a{t}_{3}^{2}$
but using equation (4)
${s}_{3}=a{t}_{1}{t}_{1}-\frac{1}{2}a{t}_{1}^{2}$
$⇒{s}_{3}=\frac{1}{2}a{t}_{1}^{2}$         ..............(5)

Now if total distance travelled by particle is s and total time taken is t then

and

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