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 16.0 m/s and initial angle of 36.8° 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?

To answer these questions, we can use basic equations of motion and kinematics. Let's break down each question step-by-step.

(a) Where will the golf ball fall back to the ground?
To determine where the golf ball will fall back to the ground, we need to find the horizontal distance it travels before hitting the ground. We can use the equation for horizontal distance traveled:
Horizontal distance = initial velocity * time * cosine of the launch angle

Given:
Initial velocity (v0) = 16.0 m/s
Initial angle (θ) = 36.8°

First, convert the angle from degrees to radians:
θ (in radians) = θ (in degrees) * π/180
θ (in radians) = 36.8° * π/180
θ (in radians) ≈ 0.6427 radians

Next, we need to find the time of flight. The time it takes for the ball to hit the ground can be determined using the equation:
Time of flight = (2 * v0 * sin θ) / g

Where:
g ≈ 9.8 m/s^2 (acceleration due to gravity)

Time of flight = (2 * 16.0 m/s * sin(0.6427 radians)) / 9.8 m/s^2
Time of flight ≈ 1.092 seconds

Now, we can calculate the horizontal distance:
Horizontal distance = 16.0 m/s * 1.092 seconds * cos(0.6427 radians)
Horizontal distance ≈ 11.66 meters

Therefore, the golf ball will fall back to the ground approximately 11.66 meters horizontally from the origin.

(b) How high will it be at the highest point of its trajectory?
To find the maximum height, we can use the equation for vertical displacement:

Vertical displacement = (v0^2 * sin^2 θ) / (2 * g)

Vertical displacement = (16.0 m/s)^2 * sin^2(0.6427 radians) / (2 * 9.8 m/s^2)
Vertical displacement ≈ 4.11 meters

Therefore, the golf ball will reach a maximum height of approximately 4.11 meters.

(c) What is the ball's velocity vector (in Cartesian components) at the highest point of its trajectory?
At the highest point of the trajectory, the vertical component of velocity is zero, while the horizontal component remains constant. Therefore, the ball's velocity vector at the highest point only has a horizontal component.

The horizontal component of the velocity can be calculated using the equation:

Horizontal velocity (Vx) = v0 * cos θ

Horizontal velocity (Vx) = 16.0 m/s * cos(0.6427 radians)
Horizontal velocity (Vx) ≈ 13.905 m/s

Therefore, the ball's velocity vector at the highest point of its trajectory is approximately 13.905 m/s in the horizontal direction only.

(d) What is the ball's acceleration vector (in Cartesian components) at the highest point of its trajectory?
At the highest point of the trajectory, the ball is at its maximum height, and the only force acting on it is gravity in the downward direction (negative y-direction). Therefore, the acceleration vector at the highest point only has a vertical component.

The vertical component of the acceleration due to gravity is -9.8 m/s^2 in the negative y-direction.

Therefore, the ball's acceleration vector at the highest point of its trajectory is approximately -9.8 m/s^2 in the negative y-direction.

To answer these questions, we can use the kinematic equations of projectile motion. Let's break down each question and go through the steps of finding the answers.

(a) Where will the golf ball fall back to the ground?
To determine where the golf ball will fall back to the ground, we can find the total time of flight and use the horizontal velocity component.

Step 1: Find the time of flight.
The time of flight can be calculated using the formula:
time of flight = (2 * initial vertical velocity) / gravitational acceleration

In this case, the initial vertical velocity is given by:
initial vertical velocity = initial velocity * sin(initial angle)

So, the time of flight can be calculated as:
time of flight = (2 * (initial velocity * sin(initial angle))) / gravitational acceleration

Step 2: Find the horizontal distance traveled.
The horizontal distance traveled can be calculated using the formula:
horizontal distance = horizontal velocity * time of flight

Since the horizontal velocity is given by:
horizontal velocity = initial velocity * cos(initial angle)

We can now determine the horizontal distance traveled by substituting the values:
horizontal distance = (initial velocity * cos(initial angle)) * time of flight

(b) How high will it be at the highest point of its trajectory?
To find the height at the highest point of the trajectory, we can determine the vertical displacement using the formula:

vertical displacement = (initial vertical velocity^2) / (2 * gravitational acceleration)

In this case, the initial vertical velocity is given by:
initial vertical velocity = initial velocity * sin(initial angle)

So, the vertical displacement can be calculated as:
vertical displacement = (initial velocity * sin(initial angle))^2 / (2 * gravitational acceleration)

(c) What is the ball's velocity vector (in Cartesian components) at the highest point of its trajectory?
At the highest point of its trajectory, the vertical velocity component becomes zero, and only the horizontal velocity component remains.
We can calculate the horizontal velocity component using the formula:
horizontal velocity component = initial velocity * cos(initial angle)

(d) What is the ball's acceleration vector (in Cartesian components) at the highest point of its trajectory?
The acceleration vector at the highest point of its trajectory only has a vertical component, which is equal to the gravitational acceleration (assuming no other forces are acting on it).

Thus, the Cartesian components of the velocity and acceleration vectors at the highest point of the trajectory are:

Velocity vector: (horizontal velocity component, 0)

Acceleration vector: (0, -gravitational acceleration)

By using these formulas and substituting the given values, you can find the answers to each part of the question.