A stone is thrown vertically upward. On its way up it passes...

A stone is thrown vertically upward. On its way up it passes point $A$ with speed $v$, and point $B, 3.00 \mathrm{~m}$ higher than $A$, with speed $\frac{1}{2} v .$ Calculate (a) the speed $v$ and (b) the maximum height reached by the stone above point $B$.

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Key Concepts

Uniformly Accelerated Motion

This concept describes motion under a constant acceleration, such as the acceleration due to gravity. It provides the framework for using kinematic equations that relate velocity, displacement, and time in situations like vertical motion, where an object’s speed decreases as it rises and increases as it falls.

Conservation of Mechanical Energy

The principle of conservation of mechanical energy states that in the absence of non-conservative forces, the total energy of a system—comprising kinetic and gravitational potential energy—remains constant throughout the motion. This allows for the comparison of speeds at different heights by equating the decrease in kinetic energy with the increase in potential energy.

Maximum Height in Vertical Motion

Maximum height is reached when the upward velocity of a projectile becomes zero. At this point, all the kinetic energy associated with upward motion has been converted into gravitational potential energy. This concept is key in determining the apex of the trajectory in vertical projectile problems.

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