Question

In the following figure all surfaces are assumed to be frictionless and pulley is assumed to be ideal. The block ${}^{\prime }{A}^{\prime }$ is projected towards the pulley ${}^{\prime }{P}^{\prime }$ with an initial velocity ${\text{u}}_0$ then select correct option:
(Assume $L_0$ is large enough so that $A$ does not strike the pulley)

• A. The string would become tight at $t=\frac{2{\text{u}}_0}{\text{g}}$
• B. The distance travelled by ${}^{\prime }{A}^{\prime }$ before the string is taut is $\frac{{\text{u}}_0^2}{\text{g}}$
• C. The distance travelled by ${}^{\prime }{B}^{\prime }$ before string is taut is $\frac{2{\text{u}}_0^2}{\text{g}}$
• D. The common speed of the blocks just after the string is taut is $\left(\begin{array}{c}\frac{\text{n+2}}{\text{n+1}}\end{array}\right){\text{u}}_0$

Answer 1

Answer: (A), (C), (D)

The correct options are
A The string would become tight at $t=\frac{2{\text{u}}_0}{\text{g}}$
C The distance travelled by ${}^{\prime }{B}^{\prime }$ before string is taut is $\frac{2{\text{u}}_0^2}{\text{g}}$
D The common speed of the blocks just after the string is taut is $\left(\begin{array}{c}\frac{\text{n+2}}{\text{n+1}}\end{array}\right){\text{u}}_0$

When the string is taut again both the block would travel the same distance.

Hence ${\text{u}}_{0}t=\frac{1}{2}{\text{gt}}^2\left[\text{as 'B' is falling freely}\right]$

Hence $\text{t}=\frac{2{\text{u}}_0}{\text{g}}\text{in this time 'A' and 'B' would travel a distance of}\frac{2{\text{u}}_0^2}{\text{g}}$

Also when the same impulse acts along the wire the change in momentum would be same for both blocks.

${\text{j = nmv - nmu}}_0\text{(for block A)}$

$-\text{J = mv - m}$ $\left(2{\text{u}}_0\right)\text{(for block B)}$

$\therefore \text{nmv}-{\text{nmu}}_0=2{\text{mu}}_0-\text{mv}$

$(\text{n + 1)v}={\text{(n + 2)u}}_0$

$\text{v}=\left(\begin{array}{c}\frac{\text{n + 2}}{\text{n + 1}}\end{array}\right){\text{u}}_0$