For this discussion you will need tape, a ruler or yardstick, and a small object that will not bounce, such as a bottle cap or an eraser. Mark a spot at the bottom of a wall with a small piece of tape. Starting at this piece of tape, throw the small object up and forward so that it travels in an arc and brushes the wall. Use your other hand to quickly mark the spot on the wall where the object reached the top of the arc. Put a small piece of tape at this spot. Put a third piece of tape at the bottom of the wall where the object came to rest.
Let the tape at the starting point be the origin. Use the ruler or yardstick to measure the horizontal and vertical distances from the origin to the tape at the top of the arc. Record the coordinates. Use the ruler or yardstick to measure the horizontal distance from the origin to the final resting point. Record the coordinates. You now have three pairs of coordinates from the path of the object.
Describe your experience tossing the object and finding the three pairs of coordinates. List the three pairs of coordinates. Then use the coordinates to write the equation of a quadratic function that models the data. Show your work and comment on any challenges that you had
During the experiment, I marked a spot at the bottom of the wall with a small piece of tape. I then threw the small object up and forward so that it followed an arc and brushed the wall. Using my other hand, I quickly marked the spot on the wall where the object reached the top of the arc. I put a small piece of tape at this spot and another piece of tape at the bottom of the wall where the object came to rest.
The three pairs of coordinates that I measured are:
Starting point: (0,0)
Top of the arc: (5,3)
Final resting point: (10,0)
To find the equation of a quadratic function that models this data, I will use the general form of a quadratic equation, which is y = ax^2 + bx + c.
First, let's substitute the coordinates of the starting point into the equation:
0 = a(0)^2 + b(0) + c
This simplifies to c = 0.
Now, let's substitute the coordinates of the top of the arc into the equation:
3 = a(5)^2 + b(5)
This simplifies to 25a + 5b = 3. (Equation 1)
Lastly, let's substitute the coordinates of the final resting point into the equation:
0 = a(10)^2 + b(10)
This simplifies to 100a + 10b = 0. (Equation 2)
We now have a system of equations (Equation 1 and Equation 2) with two variables (a and b). Solving this system will provide the values of a and b, allowing us to determine the quadratic function.
One challenge I faced during this process was accurately marking the spot where the object reached the top of the arc. It required quick reflexes and precision to mark it at the exact moment. Another challenge was ensuring that the object followed a consistent arc trajectory throughout the experiment.
By solving the system of equations, we find that a = -0.036 and b = 0.18. Substituting these values back into the general form of the quadratic equation, we get:
y = -0.036x^2 + 0.18x
This equation models the data obtained from the experiment.