Two masses are attached, each to either end of a spring. The...

Two masses are attached, each to either end of a spring. The spring is compressed till its potential energy is $\mathrm{U}$ and then released. Their velocity of separation at the instant the heavier mass has a momentum $\mathrm{p}$ is less than or equal to (a) $\frac{U}{2 p}$ (b) $\frac{\mathrm{U}}{\mathrm{p}}$ (c) $\frac{2 \mathrm{U}}{\mathrm{p}}$ (d) $\frac{4 \mathrm{U}}{\mathrm{p}}$

Two masses are attached, each to either end of a spring. The spring is compressed till its potential energy is $\mathrm{U}$ and then released. Their velocity of separation at the instant the heavier mass has a momentum $\mathrm{p}$ is less than or equal to
(a) $\frac{U}{2 p}$
(b) $\frac{\mathrm{U}}{\mathrm{p}}$
(c) $\frac{2 \mathrm{U}}{\mathrm{p}}$
(d) $\frac{4 \mathrm{U}}{\mathrm{p}}$

Key Concepts

Relation Between Kinetic Energy, Mass, and Velocity

Kinetic energy, given by ½mv², links an object's mass and velocity. Alternatively, expressing kinetic energy in terms of momentum as p²/(2m) can be useful when a problem specifies momentum constraints. This relation helps in bounding one variable when others are known.

Relative Velocity in Multi-Body Systems

Relative velocity is the speed of one object as observed from another's frame of reference. In systems where components move in opposite directions, understanding relative velocity is key to determining the overall separation speed between the bodies, particularly when dealing with momentum and energy distribution.

Conservation of Momentum

This principle states that in an isolated system, the total momentum remains constant if no external forces act on it. It is especially important in problems involving the interaction of multiple bodies where internal forces (such as those from a spring) only redistribute momentum among the bodies without changing the overall momentum.

Energy Conservation and Conversion

Energy conservation is the principle that the total energy in an isolated system remains constant. In scenarios where potential energy (like elastic potential energy stored in a compressed spring) is converted into kinetic energy, this concept allows us to set bounds on the velocities and momenta of the objects involved.

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