Daredevil Danny attempts the feat of jumping his motorcycle through the Flaming Hoop Jump of Awesome. In order for Daredevil Danny to pass through the hoop, he will need a safe path to travel. Let’s explore the parabolic trajectory that he will need for a safe journey. Directions Complete each of the following tasks, reading the directions carefully as you go. Be sure to show all work where indicated, including inserting images of graphs. Be sure that all graphs or screenshots include appropriate information such as titles, labeled axes, etc. If your word processing program has an equation editor, you can insert your equations here. Otherwise, print this activity sheet and write your answers by hand.
In addition to the answers you determine, you will be graded based on the work you show, or your solution process. So, be sure to show all your work and answer each question as you complete the task. Type all your work into this document so you can submit it to your teacher for a grade. You will be given partial credit based on the work you show and the completeness and accuracy of your explanations.
Your teacher will give you further directions about how to submit your work. You may be asked to upload the document, e-mail it to your teacher, or print it and hand in a hard copy.
Now, let’s get started!
The Correct Answer and Explanation is :
To complete the task of determining the safe path for Daredevil Danny’s motorcycle as it jumps through the Flaming Hoop Jump of Awesome, we need to analyze the parabolic trajectory of the jump. The trajectory of any object launched or projected under the influence of gravity follows a parabolic path. We will break down the process into a series of steps for clarity.
Task Overview:
- Understand the Parabolic Trajectory:
The motion of Daredevil Danny’s motorcycle is governed by the physics of projectile motion. The equation for a parabolic trajectory is:
[
y = ax^2 + bx + c
]
where ( y ) is the vertical position (height) of the motorcycle, and ( x ) is the horizontal position (distance). The constants ( a ), ( b ), and ( c ) determine the shape and position of the parabola. - Determine the Parameters for the Equation:
The specific values for ( a ), ( b ), and ( c ) will depend on the height of the starting point, the initial velocity of the jump, and the angle at which Danny launches his motorcycle. The information typically provided includes the initial velocity, launch angle, and maximum height, which will allow us to calculate the appropriate values. - Finding the Parabola’s Vertex:
The vertex of the parabola is the highest point in the jump. It occurs at the value of ( x ) where the derivative of the equation equals zero. The formula for the vertex of a parabola in standard form ( y = ax^2 + bx + c ) is:
[
x_{\text{vertex}} = \frac{-b}{2a}
]
Once you find the vertex’s ( x )-coordinate, substitute it into the equation to find the maximum height ( y_{\text{vertex}} ). - Ensuring a Safe Path Through the Hoop:
For the jump to be successful, the trajectory must pass through the hoop. This means that the motorcycle’s parabolic path must intersect with the vertical position of the hoop at the correct time. We can use the equation of the parabola to determine if the jump reaches the height of the hoop at the right horizontal distance.
Explanation of the Steps:
- Start with Basic Projectile Motion Equations:
The motion in the vertical direction is influenced by gravity, while the horizontal motion is uniform (constant velocity). The basic projectile motion equations are:
- ( y = y_0 + v_0 \sin(\theta) t – \frac{1}{2} g t^2 )
- ( x = v_0 \cos(\theta) t )
Here, ( y_0 ) is the initial height, ( v_0 ) is the initial velocity, ( \theta ) is the launch angle, and ( g ) is the acceleration due to gravity.
- Substitute Known Values:
Using the given launch velocity and angle, you can calculate the position of the motorcycle over time. From there, the parabolic trajectory can be plotted, and we can calculate when the motorcycle will reach the hoop. - Graph the Parabola:
Plot the equation of the parabola on a graph, with the ( x )-axis representing horizontal distance and the ( y )-axis representing height. Label the axes and title the graph appropriately, showing the trajectory of the motorcycle relative to the height and location of the hoop. - Ensure the Jump Is Safe:
The key is ensuring that the motorcycle reaches the appropriate height when it reaches the horizontal distance of the hoop. If the parabola intersects the hoop at the right point, then Daredevil Danny can safely jump through it.
By following these steps, you can calculate the exact trajectory and confirm that the jump is safe. The accuracy of the jump will depend on proper measurements, such as the correct initial velocity, launch angle, and other relevant factors.