Shows the speed v versus height y of a ball tossed directly...

Shows the speed $v$ versus height $y$ of a ball tossed directly upward, along a $y$ axis. Distance $d$ is $0.40 \mathrm{~m} .$
The speed at height $y_{A}$ is $v_{A} .$ The speed at height $y_{s}$ is $3 v_{A}$. What is speed $v_{A} ?$

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Key Concepts

Conservation of Mechanical Energy

This concept states that when non-conservative forces (like friction) are negligible, the total mechanical energy (the sum of kinetic and potential energies) in a system remains constant. It allows us to relate different states (for example, speeds at different heights) by equating the loss in kinetic energy to the gain in potential energy.

Potential and Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, while gravitational potential energy is the energy stored by virtue of an object's position in a gravitational field. The relationship between these energies is central to analyzing motions such as upward or downward trajectories under gravity.

Projectile Motion under Gravity

In projectile motion, an object moves under the influence of gravity, and its speed and height are interrelated through energy transformations. By understanding how energy is exchanged between kinetic and gravitational potential forms, one can determine quantities like speed at various points along the trajectory.

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