A body is projected with an initial velocity u at angle \(\theta\) to the horizontal. If \(R_{max}\) is the maximum range of the projectile, what does the r...

The relation \(\frac{u^{2}}{R_{max}}\) represents the horizontal distance covered by the projectile. When a body is projected at an angle \(\theta\) to the horizontal with an initial velocity u, it follows a curved path called a projectile motion. The path of the projectile can be divided into two components: the horizontal component and the vertical component. The horizontal component of the projectile's velocity remains constant throughout the motion because there is no acceleration acting on it in that direction. On the other hand, the vertical component of the velocity changes due to the acceleration due to gravity. The maximum range \(R_{max}\) of the projectile is the horizontal distance covered by it before hitting the ground. It can be shown that the maximum range is given by the formula: \(R_{max} = \frac{u^{2}\sin 2\theta}{g}\) where g is the acceleration due to gravity. Rearranging this equation, we get: \(\frac{u^{2}}{R_{max}} = \frac{g}{\sin 2\theta}\) Therefore, the relation \(\frac{u^{2}}{R_{max}}\) represents the inverse of the sine of twice the angle of projection multiplied by the acceleration due to gravity. This shows that the relation is a constant value for a given angle of projection and acceleration due to gravity. It represents the horizontal distance covered by the projectile, which is the maximum range.