Show that for the same initial speed \(v_{0}\) the speed \(v\) of a projectile will be the same at all points at the same elevation, regardless of the angle of projection. Ignore air drag.

When we discuss projectile motion, it's crucial to understand that in a closed system where no non-conservative forces (like friction or air resistance) are at play, the mechanical energy is conserved. This principle is known as the conservation of mechanical energy. In this context, 'mechanical energy' includes two main forms: kinetic energy—the energy of motion—and potential energy—the energy due to position.

For a projectile launched with an initial speed \(v_0\), assuming no energy is lost to air drag, the sum of kinetic and potential energy at any point in its trajectory will equal the initial kinetic energy. This is because the potential energy at the start is zero (we're considering the launch point as the datum). As the projectile rises, its kinetic energy decreases while its potential energy increases by an equal amount, preserving the total mechanical energy.

Kinetic energy (K.E.) is the energy that an object possesses due to its motion. It depends on two variables: the mass of the object \(m\) and its velocity \(v\). The mathematical formula to calculate kinetic energy is \(\frac{1}{2}mv^2\).

In the context of projectile motion, the kinetic energy varies as the projectile's velocity changes due to gravity's influence. At launch, the projectile has maximum kinetic energy because its speed is at its peak \(v_0\). As it ascends, the speed decreases, and so does the kinetic energy, reaching a minimum at the highest point in its trajectory (if vertical velocity is considered).

Potential energy (P.E.), on the other hand, is the stored energy in an object due to its position or arrangement. For objects near the Earth's surface, this is often due to their height and the force of gravity. The potential energy is given by the formula \(mgh\), where \(g\) is the acceleration due to gravity and \(h\) is the height above a chosen reference level.

In projectile motion, as the projectile rises after being launched, it slows down due to gravity pulling it downwards. This loss in speed (and therefore kinetic energy) leads to an increase in potential energy. When the projectile starts descending, the potential energy is converted back to kinetic energy. At any two points on the trajectory at the same elevation, the potential energy will be the same, contributing to the conservation of mechanical energy.

The angle of projection is the angle at which a projectile is launched with respect to the horizontal. It substantially affects the motion's initial horizontal and vertical velocity components. Varying the angle while keeping the initial speed \(v_0\) constant will alter the range, height, and duration of the flight, but not the magnitude of the speed at equal elevations.

For instance, an object projected at a 45-degree angle will reach a different maximum height than one at 30 degrees, but if you measure their speed at the same height on their upward or downward path, conservation of mechanical energy tells us the speed will be identical. This uniformity in speed at the same height, regardless of the angle, is a fascinating consequence of energy conservation in projectile motion.