A small body A starts sliding from the height h down an inclined groove passing into a half-circle of radius `h//2` (figure).

Assuming the friction to be negligible, find the velocity of the body at the highest point of its trajectory (after breaking off the groove).

A body A is dropped from a height h above the ground. At the same time another body B (at a horizontal' distance d from the line of fall of A) is fired from the ground at an angle alpha to the horizontal. If the two bodies collide at the highest point of the trajectory of B the angle of projection is given by

A body slides down a fixed curved track that is one quadrant of a circle of radius R , as in the figure. If there is no friction and the body starts from rest, its speed at the bottom of the track is

A small body of mass m slide without friction from the top of a hemispherical cup of radius r as shown in figure. If leaves the surface to the cup at vertial distance h below the highest point then

A small body of mass m slides down from the top of a frictionless hemispherical surface of radius r as shown in the figure.The vertical height h below the highest point at which the body gets separated from the hemispherical surface. .

In the system shown in figure the masses of the bodies are known to be m_1 and m_2 , the coefficient of friction between the body m_1 and the horizontal plane is equal to k , and a pulley of mass m is assumed to be a uniform disc. The thread does not slip over the pulley. At the moment t=0 the body m_2 starts descending. Assuming the mass of the thread and the friction in the axle of the pulley to be negligible, find the work performed by the friction forces acting on the body m_1 over the first t seconds after the beginning of motion.