Three bodies are thrown vertically up simultaneously as...

Three bodies are thrown vertically up simultaneously as shown in the figure, so that all three collide. Then (a) $\mathrm{u}_{2}=\frac{\mathrm{u}_{1}+\mathrm{u}_{3}}{2}$ (b) $\mathrm{u}_{2}=\sqrt{\mathrm{u}_{1} \mathrm{u}_{3}}$ (c) $\mathrm{u}_{2}=\sqrt{\frac{\mathrm{u}_{1}^{2}+\mathrm{u}_{3}^{2}}{2}}$ (d) $\frac{2}{\mathrm{u}_{2}}=\frac{1}{\mathrm{u}_{1}}+\frac{1}{\mathrm{u}_{3}}$

Three bodies are thrown vertically up simultaneously as shown in the figure, so that all three collide. Then
(a) $\mathrm{u}_{2}=\frac{\mathrm{u}_{1}+\mathrm{u}_{3}}{2}$
(b) $\mathrm{u}_{2}=\sqrt{\mathrm{u}_{1} \mathrm{u}_{3}}$
(c) $\mathrm{u}_{2}=\sqrt{\frac{\mathrm{u}_{1}^{2}+\mathrm{u}_{3}^{2}}{2}}$
(d) $\frac{2}{\mathrm{u}_{2}}=\frac{1}{\mathrm{u}_{1}}+\frac{1}{\mathrm{u}_{3}}$

Key Concepts

Mean Value Relationships in Physics

Mean value relationships, including arithmetic, geometric, harmonic, and quadratic means, are mathematical tools used to describe average values of physical quantities. In problems involving simultaneous collisions or averaging velocities, these means help relate the speeds or energies of different objects to satisfy certain physical criteria.

Relative Motion and Collision Conditions

Relative motion pertains to the movement of one object as observed from another moving reference frame, which is important when determining when and where multiple objects collide. Setting the time of collision equal for all bodies leads to relationships between their initial velocities, providing insight into the conditions needed for simultaneous collision.

Projectile Motion

Projectile motion involves the motion of an object thrown into a gravitational field, where it moves along a curved path under the influence of gravity alone. This concept covers fundamental kinematic equations that relate displacement, velocity, acceleration, and time, and is essential for analyzing any vertical motion scenario where bodies are thrown upward or downward.

Uniform Acceleration Kinematics

Uniform acceleration kinematics deals with the motion of objects under a constant acceleration, such as gravity. It provides the mathematical framework to determine parameters like displacement, velocity, and time, which are crucial when analyzing collisions or meeting conditions in vertical motions.

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