Question

For a particle undergoing rectilinear motion with uniform acceleration, the magnitude of displacement is
one third the distance covered in some time interval. The magnitude of final velocity is less than
magnitude of initial velocity for this time interval. Then the ratio of initial speed to the final speed for this
time interval is :
(A) V2
(B) 2
(C) 73
(D) 3​

Answer 1

Answer:

$\sqrt{2}$

Step-by-step explanation:

1) Let the initial position of particle be O with some velocity 'u'.

According to story given in question, particle has some negative acceleration/retardation .(towards left).

Then , its velocity becomes 0 at B at t=t_b.

Again, due to acceleration it reaches A at time t_a with some velocity 'v'.

2) Now,

Since, displacement  over journey is one third of distance covered.

That is, OA=OB=d(say)

As OB+BA= 1/3 of OA

Using Newtons  Equation,

From O to B,

$v^2-u^2=2as\\ \\=>0^2-u^2=2a(2d)\\ \\=>a=\frac{-u^2}{4d}$

3) From B to A,

Using Newtons Equation,

$v^2-u^2=2as\\ \\=>v^2-0=2*\frac{-u^2}{4d}*(-d)\\ \\=>\frac{u^2}{v^2} =2\\ \\=>\frac{u}{v}=\sqrt{2}$

Hence, Required answer is

$\boxed{\sqrt2}$

Answer 2

Step-by-step explanation:

1) Let the initial position of particle be O with some velocity 'u'.

According to story given in question, particle has some negative acceleration/retardation .(towards left).

Then , its velocity becomes 0 at B at t=t_b.

Again, due to acceleration it reaches A at time t_a with some velocity 'v'.

2) Now,

Since, displacement over journey is one third of distance covered.

That is, OA=OB=d(say)

As OB+BA= 1/3 of OA

Using Newtons Equation,

From O to B,

\begin{gathered}v^2-u^2=2as\\ \\= > 0^2-u^2=2a(2d)\\ \\= > a=\frac{-u^2}{4d}\end{gathered}

v

2

−u

2

=2as

=>0

2

−u

2

=2a(2d)

=>a=

4d

−u

2

3) From B to A,

Using Newtons Equation,

\begin{gathered}v^2-u^2=2as\\ \\= > v^2-0=2*\frac{-u^2}{4d}*(-d)\\ \\= > \frac{u^2}{v^2} =2\\ \\= > \frac{u}{v}=\sqrt{2}\end{gathered}

v

2

−u

2

=2as

=>v

2

−0=2∗

4d

−u

2

∗(−d)

=>

v

2

u

2

=2

=>

v

u

=

2

Hence, Required answer is

2^1/2