Consider two projectiles launched on level ground with the same speed, at angles \(45^{\circ} \pm \alpha .\) Show that the ratio of their flight times is \(\tan \left(\alpha+45^{\circ}\right)\)

Projectile motion refers to the motion of an object that is thrown or projected into the air, subject to only the acceleration of gravity. The path of a projectile is a parabola under ideal conditions. Understanding projectile motion is critical for predicting the behavior of objects launched at an angle.

For any projectile, the horizontal and vertical components of its velocity can be analyzed separately. The horizontal component \( v_x \) remains constant, as there is no acceleration in the horizontal direction (neglecting air resistance), while the vertical component \( v_y \) changes due to acceleration caused by gravity \( g \).

In the context of our exercise, we consider two projectiles launched at different angles but with the same initial speed. This has a direct effect on their flight times and trajectories, as the initial vertical and horizontal components of speed attributed to each angle will differ. As a result, this will alter their time of flight, which is the duration the projectile remains in the air. The analysis of projectile motion allows us to derive the equation that determines the ratio of their flight times.

The angle of launch, or launch angle, plays a pivotal role in projectile motion. It is the angle at which an object is projected with respect to the horizontal. The optimal angle for distance in a vacuum, or without air resistance, is \(45^{\text{o}}\) because it provides the perfect balance between the vertical and horizontal components of the initial velocity.

When two projectiles are launched with the same initial speed but at different launch angles, their flight paths and times of flight will differ. A projectile launched at an angle greater than \(45^{\text{o}}\) will have a longer air time but may not cover as much horizontal distance, while a projectile launched at an angle less than \(45^{\text{o}}\) may travel a greater horizontal distance but will have a shorter air time.

The exercise we're delving into examines two angles symmetrical about \(45^{\text{o}}\), specifically \(45^{\text{o}} + \alpha\) and \(45^{\text{o}} - \alpha\). This symmetry about the optimal angle of \(45^{\text{o}}\) allows for a comparison that yields straightforward mathematical relations between the flight times of the two projectiles.

Time of flight in projectile motion is the total time that the projectile is in the air. For projectiles without air resistance and launched from level ground, this time can be calculated using the formula \( T = \dfrac{2u\sin(\theta)}{g} \), where \( T \) is the time of flight, \( u \) is the initial velocity, \( \theta \) is the launch angle, and \( g \) is the acceleration due to gravity. It is derived from the motion in the vertical direction and the fact that the final vertical velocity at the moment just before impact is equal in magnitude and opposite in direction to the initial vertical velocity.

As demonstrated in the solution steps for our exercise, by calculating the time of flight for each angle, we're able to find a ratio that compares the two flight times. This ratio provides a neat conceptual understanding of how different launch angles relative to \(45^{\text{o}}\) affect the projectile's air time and is a practical example of how trigonometric identities are applied in physics problems.

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables where both sides of the equation are defined. These identities are useful for simplifying trigonometric expressions and solving trigonometric equations.

One commonly used set of identities is the angle sum and difference identities. For our particular exercise, the sine of an angle sum \( \sin(\alpha + \beta) \) and angle difference \( \sin(\alpha - \beta) \) are used to simplify the ratio of the times of flight for projectiles launched at \(45^{\text{o}} + \alpha\) and \(45^{\text{o}} - \alpha\).

Specifically, \( \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) \) and \( \sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \cos(\alpha)\sin(\beta) \). Using these identities, the ratio of the sines in the flight time ratio simplifies to the tangent of the sum \( \tan(\alpha + 45^{\text{o}}) \) which neatly encapsulates the relationship between the launch angles and their corresponding flight times.