For a science fair competition, a group of high school students build a kicker-machine that can launch a golf ball from the origin with a velocity of \(11.2 \mathrm{~m} / \mathrm{s}\) and initial angle of \(31.5^{\circ}\) with respect to the horizontal. a) Where will the golf ball fall back to the ground? b) How high will it be at the highest point of its trajectory? c) What is the ball's velocity vector (in Cartesian components) at the highest point of its trajectory? d) What is the ball's acceleration vector (in Cartesian components) at the highest point of its trajectory?

When studying projectile motion, we often use kinematic equations to describe the motion of the object in terms of its position, velocity, and acceleration over time. These equations are the tools that allow us to predict future movements from current measurements.

Specifically, in projectile motion, we have two components to consider: horizontal and vertical. The horizontal motion is uniform, meaning that there is no acceleration, while the vertical motion is uniformly accelerated due to gravity. The kinematic equations for vertical motion, which include terms for initial velocity (\( V_{0y} \text{or}\ V_0\text{sin}\theta \text{where}\theta \text{is the launch angle} \)), time (\( t \text{how long the projectile is in motion} \)), acceleration due to gravity (\( g \)), and vertical position (\( y \)), provide the basis for solving many problems related to projectile motion.

The trajectory of a projectile is the path it follows under the influence of gravity alone. After being projected, the object follows a curved path known as a parabola before returning to the ground. This parabolic trajectory is influenced by the initial speed and the angle of launch.

With an angle of elevation of \(31.5^\text{\circ}\) and an initial speed, our golf ball moves out and up, then down and out, tracing a symmetric path. The trajectory is determined by both the vertical and horizontal components of motion, with gravity affecting only the vertical component. The analysis of projectile trajectory allows us to calculate the maximum height the object reaches and the distance it travels horizontally before touching the ground.

The time of flight in projectile motion refers to the total duration the projectile remains airborne. For our golf ball, this is the time from launch until it returns to the ground. It is directly related to the vertical component of the initial velocity and the acceleration due to gravity.

To calculate the time of flight, we consider the moment when the golf ball lands (\( y = 0 \)) and solve for \( t \) considering upward motion (positive) and downward motion (negative due to gravity). The equation \( 0 = V_0\text{sin}(\theta)t - 0.5gt^2 \) helps us find that time, and once it's known, it allows us to calculate other properties, like the horizontal displacement of the projectile.

The maximum height a projectile achieves occurs at the apex of its trajectory, where the vertical component of velocity momentarily becomes zero (\( V_{y} = 0 \)). At this point, the only component of velocity that remains is the horizontal one, as it is unaffected by gravity.

Using the equation \( V_{y}^2 = V_{0y}^2 - 2gy \), and setting \( V_{y} = 0 \) helps to isolate for \( y \) giving us the maximum height. This equation reveals the direct relationship between the initial speed of the projectile, the angle of projection, and the peak height it can reach in the absence of any other forces like air resistance.

The velocity of a projectile is composed of both horizontal and vertical velocity vector components. At launch, the velocity will have both a positive vertical component and a horizontal component, based on the projection angle. However, at the peak of its flight, the vertical component becomes zero (\( V_{y} = 0 \)) while the horizontal component remains constant due to the absence of horizontal forces in ideal projectile motion.

The constant horizontal velocity component (\( V_{0x} \)) can be calculated using \( V_{0x} = V_0\text{cos}(\theta) \). These vector components allow us to understand the motion of the projectile in more detail at any point in its trajectory.

The acceleration due to gravity is a constant value (\(9.8 \text{m/s}^2\) on the surface of Earth) that acts on all objects equally, pulling them towards the Earth's center. This acceleration is always directed downward in vertical projectile motion problems.

In the absence of air resistance, the only acceleration affecting our golf ball is the acceleration due to gravity, hence why its vertical component of acceleration is \(-9.8 \text{m/s}^2\), which is constant throughout the motion, while the horizontal component of acceleration is zero.