Two balls $$A$$ and $$B$$, of masses $$M$$ and $$2 M$$ respectively collide each other. If the ball $$A$$ moves with a speed of $$150 \mathrm{~ms}^{-1}$$ and collides with ball $$B$$, moving with speed $$v$$ in the opposite direction. After collision if ball $$A$$ comes to rest and the coefficient of restitution is 1 (one), then the speed of the ball $$B$$ before it collides with ball $$A$$ is

As shown in the figure, an iron block $$A$$ of volume $$0.25 \mathrm{~m}^3$$ is attached to a spring $$S$$ of unstretched length 1.0 m and hanging to the ceiling of a roof. The spring gets stretched by 0.2 m . This block is removed and another block $$B$$ of iron of volume $$0.75 \mathrm{~m}^3$$ is now attached to the same spring and kept on a frictionless incline plane of $$30^{\circ}$$ inclination. The distance of the block from the top along the incline at equilibrium is

A ball of mass 0.5 kg moving horizontally at $$10 \mathrm{~ms}^{-1}$$ strikes a vertical wall and rebounds with speed $$v$$. The magnitude of the change in linear momentum is found to be $$8.0 \mathrm{~kg}-\mathrm{~ms}^{-1}$$. The magnitude of $$v$$ is

Masses $$m\left(\frac{1}{3}\right)^N \frac{1}{N}$$ are placed at $$x=N$$, when $$N=2,3,4 \ldots \infty$$. If the total mass of the system is $$M$$, then the centre of mass is