Question

The engineering club has built a catapult and wants to test it out. The local supermarket has donated some overripe fruits and vegetables, and now the club is holding a Splatapult challenge to see who can hit the most targets. The catapult can launch food pretty far and make a real mess! In order to hit a target and win the Splatapult challenge, you'll need to aim the catapult just right. Use the Graphing Tool and your knowledge of quadratic functions to help you model the flight paths of the different projectiles and hit the target.

1. Which fruit or vegetable did you select? List the information you know about its path.

2. Your fruit or vegetable will follow a parabolic path, where x is the horizontal distance it travels (feet), and y is the vertical distance (feet).
a) The x-intercepts are the places where your fruit or vegetable is on the ground.
The first x-intercept is (0, 0).
The second x-intercept is where the fruit or vegetable hits the ground after it's launched.
What are the coordinates of the second x-intercept?
b) Which point on the parabola shows the maximum height of your fruit or vegetable?
c) The x-coordinate of the vertex is halfway between the x-coordinates of the endpoints. The y-value of the vertex is the maximum height of the fruit or vegetable. What are the coordinates of the vertex?

3. Use the information above to sketch the flight path of your projectile.

4. Label the x-intercepts and vertex on your sketch.

Complete questions 5 – 8 to write a quadratic equation for the parabola.
5. Using the coordinates of the x-intercepts, what are the two roots of your quadratic equation? (1 point: ½ point for each root)
Hint: The roots are the same as the zeros (the x-values of the x-intercepts).
r1 = _____ r2 = _____

6. Substitute the roots from question 5 into the equation y = a(x – r1)(x – r2).

7. Substitute the coordinates of the vertex for x and y in the equation from question 6. Solve the equation for a.

8. Using the value of a from question 7, write the quadratic equation in the form y = a(x – r1)(x – r2).

9. Using the distributive property, multiply the equation in question 8 to get the quadratic equation in the form y = ax2 + bx.

10. Use the Graphing Tool to create a parabola with your vertex and x-intercepts.
To use the graphing tool, first zoom out to get the scale you want.
Select the parabola, "U", button.
The click the point that you want as the vertex.
Then click the origin (0, 0).
The Graphing Tool will give you an equation for the function you have created. This equation should be close to the equation from question 8. Write the equation here:

Answer 1

The trajectory of a projectile follows a parabolic path and can be mathematically derived to form a quadratic equation. This equation is fundamental in analyzing projectile motion characteristics such as maximum height and range in physics, especially relevant in AP courses.

Step-by-step explanation:

Projectile Motion Trajectory Analysis The trajectory of a projectile is indeed parabolic, which can be demonstrated mathematically. Starting with the equations for horizontal and vertical motion, x = Voxt and y = Voyt - (1/2)gt2, where Vox and Voy are the initial velocities in the x and y components respectively, and g is the acceleration due to gravity, we can solve for t from the first equation and substitute it into the second equation. Doing so, we find that y can be expressed as a function of x, taking the form y = ax + bx2, where a and b are constants dependent upon the initial velocities and acceleration due to gravity.For a launcher propelling a projectile at an angle ϴ, the vertical height y, according to experimental data, can sometimes be represented by y = KV2 sin ϴ.

To determine if this equation is correct, we would examine if it concurs with our understanding that a launcher inclined at a greater angle (with the same initial velocity) should project a ball to a higher peak height before it falls to the ground. If this equation matches empirical data, it can be deemed consistent.The quadratic equation derivation of the projectile's path is essential in analyzing its maximum height, range, and overall trajectory. This forms a critical part of the physics secion for AP courses, aiding students in understanding the principles of independence of motion and its effects on projectile motion problems.