The potential energy in joules of a particle of mass $1\, kg$ moving in a plane is given by $U = 3x + 4y$,where the position coordinates $x$ and $y$ are measured in metres. If the particle is initially at rest at $(6, 4)$,then:

$A$ block of mass $m$ starts at rest at height $h$ on a frictionless inclined plane. The block slides down the plane,travels across a rough horizontal surface with coefficient of kinetic friction $\mu$,and compresses a spring with force constant $k$ a distance $x$ before momentarily coming to rest. Then the spring extends and the block travels back across the rough surface,sliding up the plane. The block travels a total distance $d$ on the rough horizontal surface. The correct expression for the maximum height $h'$ that the block reaches on its return is

Two masses $m_1$ and $m_2$ are connected by a string of length $l$. They are held in a horizontal plane at a height $H$ above two heavy plates $A$ and $B$ made of different materials placed on the floor. Initially,the distance between the two masses is $a < l$. When the masses are released under gravity,they collide with $A$ and $B$ with coefficients of restitution $e_1 = 0.8$ and $e_2 = 0.4$ respectively. Find the time after the collision when the string becomes tight. (Assume $H >> l$)

$A$ truck pulls a mass of $1200 \ kg$ at a constant speed of $10 \ m/s$ on a level road. The tension in the coupling is $1000 \ N$. The power used on the mass will be ..... . When the truck moves on an incline of $1 \ m$ rise in $6 \ m$ length,the tension will be ..... .

In two separate collisions,the coefficients of restitution $e_1$ and $e_2$ are in the ratio $3: 1$. In the first collision,the relative velocity of approach is twice the relative velocity of separation. Then,the ratio between the relative velocity of approach and the relative velocity of separation in the second collision is:

Consider a frictionless ramp on which a smooth object is made to slide down from an initial height $h$. The distance $d$ necessary to stop the object on a flat track (of coefficient of friction $\mu$), kept at the ramp end, is: