Two buses P and Q start from a point at the same time and move in a straight line and their positions are represented by $${X_P}(t) = \alpha t + \beta {t^2}$$ and $${X_Q}(t) = ft - {t^2}$$. At what time, both the buses have same velocity?

A disc with a flat small bottom beaker placed on it at a distance R from its center is revolving about an axis passing through the center and perpendicular to its plane with an angular velocity $$\omega$$. The coefficient of static friction between the bottom of the beaker and the surface of the disc is $$\mu$$. The beaker will revolve with the disc if :

A solid metallic cube having total surface area 24 m² is uniformly heated. If its temperature is increased by 10$$^\circ$$C, calculate the increase in volume of the cube. (Given $$\alpha$$ = 5.0 $$\times$$ 10^$$-$$4 $$^\circ$$C^$$-$$1).

A copper block of mass 5.0 kg is heated to a temperature of 500$$^\circ$$C and is placed on a large ice block. What is the maximum amount of ice that can melt? [Specific heat of copper : 0.39 J g^$$-$$1 $$^\circ$$C^$$-$$1 and latent heat of fusion of water : 335 J g^$$-$$1]