A mass of $$M$$ $$kg$$ is suspended by a weightless string. The horizontal force that is required to displace it until the string makes an angle of $${45^ \circ }$$ with the initial vertical direction is

The potential energy of a $$1$$ $$kg$$ particle free to move along the $$x$$-axis is given by $$V\left( x \right) = \left( {{{{x^4}} \over 4} - {{{x^2}} \over 2}} \right)J$$.

The total mechanical energy of the particle is $$2J.$$ Then, the maximum speed (in $$m/s$$) is

A bullet fired into a fixed target loses half of its velocity after penetrating $$3$$ $$cm.$$ How much further it will penetrate before coming to rest assuming that it faces constant resistance to motion?

A spherical ball of mass $$20$$ $$kg$$ is stationary at the top of a hill of height $$100$$ $$m$$. It rolls down a smooth surface to the ground, then climbs up another hill of height $$30$$ $$m$$ and finally rolls down to a horizontal base at a height of $$20$$ $$m$$ above the ground. The velocity attained by the ball is