91Ó°ÊÓ

In projectile motion, the concept of maximum height is pivotal. The maximum height attained by a projectile is reached when the vertical component of its velocity becomes zero. The formula governing this is given by \( H = \frac{u^2 \sin^2 \theta}{2g} \), where \( u \) is the initial velocity, \( \theta \) is the angle of projection, and \( g \) is the acceleration due to gravity. This formula shows us that:

• The maximum height is directly dependent on the square of the initial velocity and the sine of the angle squared.
• Any increase in either the initial velocity or the angle, with a constant velocity, results in a higher maximum height.

Understanding this concept aids in analyzing how changes in initial conditions affect the motion of the projectile. For instance, a change in maximum height implies a change in the velocity or the angle of projection, which consequently affects other elements such as range and time of flight.

The horizontal range in a projectile's motion is the total horizontal distance it covers before hitting the ground. This is defined by the formula \( R = \frac{u^2 \sin(2\theta)}{g} \). What makes the horizontal range intriguing is that:

• It depends on the square of the initial velocity and the sine of twice the angle of projection.
• Unlike maximum height, doubling the angle in the sine function introduces a unique dependency on the angle.

When the maximum height increases, as shown in the original exercise, it indicates an increase in the initial velocity. Retaining the same angle, yet increasing initial velocity, results in a proportional increase in range. Therefore, if maximum height increases by 5%, the horizontal range also follows this pattern and increases by 5%, emphasizing the dynamic relationship between height and range.

The time of flight is crucial in projectile motion. It represents the total time interval for which the projectile remains airborne. For horizontal motion with initial velocity \( u \) and projection angle \( \theta \), the time of flight is determined by:\[ T = \frac{2u \sin \theta}{g} \]How does an increase in height relate to the time of flight? If the maximum height increases by 10%, there is a corresponding increase in initial velocity. The revised expression for time of flight becomes:\[ T' = \frac{2(\sqrt{1.1}u) \sin \theta}{g} \]Simplifying, we find \( T' \) approximates to \( \sqrt{1.1}T \), equating to about a 4.88% increase in time of flight. This demonstrates that time of flight is not solely dependent on height, but also on the interplay between velocity and angle, which modulates how long the projectile is airborne.