The centres of the spheres 1, 2 and 3 lie on a single straight line. Sphere 1 is moving with an (initial) velocity v1 directed along this line and hits sphere 2. Sphere 2, acquiring after collision a velocity v2, hits sphere 3. Both collisions are absolutely elastic. What must be the mass of sphere 2 (in kg) for the sphere 3 to acquire maximum velocity (The masses m1 and m3, of spheres 1 and 3 are 9 kg and 1 kg respectively)?

m1v1=m1v1'+m2v2 and, e=v2-v1'v1-0=1 ⇒ v2=2m1m1+m2v1∴ v3=2m2m2+m3v2=4m1m2v1m1+m2m2+m3For maximum v3 , dv3dm2=0⇒m2=m1m3=9×1 =3 kg

The centres of the spheres 1, 2 and 3 lie on a single straight line. Sphere 1 is moving with an (initial) velocity v1 directed along this line and hits sphere 2. Sphere 2, acquiring after collision a velocity v2, hits sphere 3. Both collisions are absolutely elastic. What must be the mass of sphere 2 (in kg) for the sphere 3 to acquire maximum velocity (The masses m1 and m3, of spheres 1 and 3 are 9 kg and 1 kg respectively)?

m1v1=m1v1'+m2v2 and, e=v2-v1'v1-0=1 ⇒ v2=2m1m1+m2v1∴ v3=2m2m2+m3v2=4m1m2v1m1+m2m2+m3For maximum v3 , dv3dm2=0⇒m2=m1m3=9×1 =3 kg

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The centres of the spheres 1, 2 and 3 lie on a single straight line. Sphere 1 is moving with an (initial) velocity v₁ directed along this line and hits sphere 2. Sphere 2, acquiring after collision a velocity v₂, hits sphere 3. Both collisions are absolutely elastic. What must be the mass of sphere 2 (in kg) for the sphere 3 to acquire maximum velocity (The masses m₁ and m₃, of spheres 1 and 3 are 9 kg and 1 kg respectively)?

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5 kg

7 kg

1 kg

3 kg

answer is B.

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Detailed Solution

$m_{1} v_{1} = m_{1} v_{1}' + m_{2} v_{2} and, e = \frac{v_{2} - v_{1}'}{v_{1} - 0} = 1$

$\begin{array}{l} \Rightarrow v_{2} = \frac{2 m_{1}}{m_{1} + m_{2}} v_{1} \\ ∴ v_{3} = \frac{2 m_{2}}{m_{2} + m_{3}} v_{2} = \frac{4 m_{1} m_{2} v_{1}}{(m_{1} + m_{2}) (m_{2} + m_{3})} \end{array}$
For maximum v₃ , $\frac{{dv}_{3}}{{dm}_{2}} = 0$
$\Rightarrow m_{2} = \sqrt{m_{1} m_{3}} = \sqrt{9 \times 1} = 3 kg$