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].
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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
.................(1)
and the velocity of the particle at the end of the time interval is given by
..................(2)
2) Uniform motion : Let s2 be the distance covered by the particle in t2 time-interval with uniform velocity v so
...................(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 t3 is v and final velocity is 0 so
..................(4)
and s3 is given by
but using equation (4)
..............(5)
Now if total distance travelled by particle is s and total time taken is t then
and
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
.................(1)
and the velocity of the particle at the end of the time interval is given by
..................(2)
2) Uniform motion : Let s2 be the distance covered by the particle in t2 time-interval with uniform velocity v so
...................(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 t3 is v and final velocity is 0 so
..................(4)
and s3 is given by
but using equation (4)
..............(5)
Now if total distance travelled by particle is s and total time taken is t then
and