91Ó°ÊÓ

In projectile motion, the horizontal motion is uniform. This means the horizontal velocity remains constant throughout the flight because there are no horizontal forces acting on the projectile (neglecting air resistance). In the exercise, the tennis ball is served with a horizontal velocity of 23.6 m/s. To find the time it takes for the ball to reach the net, you can use the formula for time, which is: \ \ \( t = \frac{d}{v} \) \ \ Here, \(d = 12\) m (distance to the net) and \(v = 23.6\) m/s (horizontal speed). So, the time \( t \approx 0.508 \) s.

Vertical motion in projectile problems is influenced by gravity. The vertical velocity of the projectile changes over time because of the acceleration due to gravity (\( g = 9.8 \) m/s²). In our problem, initially, the ball was served horizontally, so its vertical velocity \( v_{y0} \) is 0. We can calculate the vertical position of the ball using the formula: \ \ \( y = y_0 + v_{y0} t - \frac{1}{2} g t^2 \) \ \ Where \( y_0 = 2.37 \) m (initial height), \( t = 0.508 \) s (time to reach the net), and \( g = 9.8 \) m/s². Plugging in these values: \ \ \( y \approx 2.37 \text{ m} - 1.27 \text{ m} \approx 1.10 \text{ m} \). This position is compared with the net's height to determine if the ball clears it.

Kinematics deals with the motion of objects without considering the forces causing the motion. The key kinematic equations help us describe the projectile's horizontal and vertical components. For horizontal motion, the time to reach a certain distance is \( t = \frac{d}{v} \) and for vertical motion, we use: \ \ \( y = y_0 + v_{y0} t - \frac{1}{2} g t^2 \) \ \ These equations allow us to solve problems involving projectiles by separating the motion into horizontal and vertical components. The solution to the tennis problem involves these basics to calculate time and heights.

Gravity is the force that pulls objects toward the Earth, giving them a constant acceleration of \( 9.8 \) m/s² downward. This force solely influences the vertical motion of a projectile. In the exercise, gravity affects the tennis ball's drop as it travels forward. Initially, the ball serves horizontally, thus vertical speed \( v_{y0} = 0 \). As time progresses, gravity causes the ball to accelerate downwards, and its position decreases based on the equation: \ \ \( y = y_0 - \frac{1}{2} g t^2 \). To compute this accurately is the key to predicting the ball's exact vertical position at any time.

In projectile problems, the initial velocity can be broken down into horizontal (\( v_{x0} \)) and vertical (\( v_{y0} \)) components. For the tennis ball, the initial speed was \( 23.6 \) m/s. When the ball is served horizontally, \( v_{y0} = 0 \). \ \ But, when the ball is served at a 5-degree angle downward, it has both horizontal and vertical components. The initial vertical velocity component is calculated using: \ \ \( v_{y0} = v \times \tan(\theta) \ \text{where V = 23.6 m/s and } \theta = 5° \). This results in \( v_{y0} = -2.06 \text{ m/s} \). Now our initial condition changes, influencing both the vertical position and trajectory. Using this, we determine the impact of the angle on clearing the net.