A golf ball is hit with an initial angle of \(35.5^{\circ}\) with respect to the horizontal and an initial velocity of \(83.3 \mathrm{mph}\). It lands a distance of \(86.8 \mathrm{~m}\) away from where it was hit. \(\mathrm{By}\) how much did the effects of wind resistance, spin, and so forth reduce the range of the golf ball from the ideal value?

Kinematics is the branch of physics that describes the motion of objects without necessarily looking at the forces that cause the motion. In the context of projectile motion, kinematics focuses on the trajectory of an object as it moves under the influence of gravity.

In projectile motion, there are two components of motion to consider: horizontal and vertical. The horizontal motion is constant because in an ideal scenario without air resistance, there is no horizontal force acting on the projectile after it's been launched. This means we can easily predict where the projectile will land horizontally by calculating the horizontal velocity multiplied by the time of flight.

However, the vertical motion is affected by gravity, which causes a projectile to accelerate downwards. We use kinematic equations to predict the vertical position and velocity at any point in time. For example, the time of flight can be calculated using the equation \( t_{flight} = \dfrac{2V_y}{g} \), where \( V_y \) is the vertical velocity, and \( g \) is the acceleration due to gravity. \( g \) is approximately \( 9.81 \text{m/s}^2 \) on Earth.

The physics of sports often involves analyzing projectile motion to improve performance and understand the effects of various factors like resistance or spin. For example, when a golf ball is hit, its trajectory is determined by the initial velocity and angle of launch. This is directly connected to the player's technique and the equipment used.

The ideal projectile motion is calculated assuming no external forces like air resistance or wind. However, in real-world scenarios, resistance, and factors such as the ball's spin, can greatly affect the ball's flight path and range. These are non-ideal conditions and are difficult to account for because of their complex and variable nature.

Therefore, understanding the basic principles of projectile motion can help athletes and coaches devise strategies to counteract these effects. For instance, in golf, players may choose particular clubs to adjust the initial velocity and angle, or they might account for wind when aiming.

Every projectile has two main velocity components: horizontal (\( V_x \) and vertical (\( V_y \). These components are determined by the initial launch velocity and the angle of launch. The horizontal velocity affects how far the projectile will travel, while the vertical velocity affects how high it will go.

Horizontal velocity is calculated using the cosine of the launch angle: \( V_x = V \cdot \cos(\theta) \), where \( V \) is the initial velocity and \( \theta \) is the launch angle. Since there are no horizontal forces acting on the projectile in ideal conditions, this component remains constant throughout the flight.

Vertical velocity, on the other hand, is calculated using the sine of the launch angle: \( V_y = V \cdot \sin(\theta) \). Unlike horizontal velocity, vertical velocity changes over time due to the acceleration of gravity, reaching zero at the peak of the trajectory before reversing direction.