Fireworks portfolio task one conduct some research to help you with later portions of the portfolio assignment find a local building an estimated height. How tall do you think the building is use any building in your area or some building that you know estimate the height of the building. Find some initial velocity’s for different types of fireworks. What are some of the initial velocity that you found do some research on the topic and find some initial velocity to fireworks attached to my setting up a fireworks display you have a tool on at the top of the building in need to drop it to eight Coworker below. How long will it take to to to fall to the ground? Use the first equation given above H parentheses T equals -16 T squared plus h naught. for the buildings height use the height of the building that you estimated in task one. H naught represents the initial height of the tool. Let h(t) be zero to represent the height being zero. solve for t (time) to know how long it will take for the tool to reach the ground. Show your work. draw a graph, the represents the path of this tool falling to the ground. Be sure to label your access with a title and a scale your graph to show the height of the tool h after t seconds have passed label this line tool. State whether the para bola represented by h(t) equals -16t squared +250topens up or down explain why your answer makes sense in the context of this problem consider what the coefficient of T squared is and how that affects the parabola. One of the fireworks launch from the top of the building with an initial upward velocity of 150 ft./s. What is the equation for the situation your side of your building from task one the initial velocity takes the place of the coefficient of t in the equation. When will the firework land? If it does not explode what is the height of the firework when it lands on the ground plug that in for height and saw for T you were looking for an X intercept you must show your work make a table for the situation so that it shows the height from time t= 0 until it hits the ground use random time values in between the time being zero in the time when it hits the ground to complete your table, calculate the access of symmetry use the formula for finding the axis of symmetry you must show your work, calculate the coordinates of the vertex use the value of the access of symmetry to find the Y coordinates of the vertex. You must show your work. explain why negative values for t and h(t) do not make sense for this problem. Think about what t and h(t) represent in the context of the scenario, and why they could not be negative. On the same coordinate plane from number one draw a graph that represents the path of this firework. Make sure your graph is labeled appropriately. Label this graph firework number one. Choose an initial velocity for a firework based on your research from task one write an equation that represents the path of a firework that is launch from the ground with the initial velocity that you choose. Choose whatever initial velocity, and consider where that fits into the equation, this time you are launching the firework from the ground so consider how that changes the equation. Suppose the firework is set to explode three seconds after it’s launched. At what height will this firework be when it explodes? Is the fireworks explode after three seconds, consider where that value of three fit into the equation to find the height of the firework at that time. you must show your work. On the same coordinate plane that you have been using draw a graph that represents the path of this fire work. Mark your graph to indicate the point at which the firework will explode. Label this graph firework number two . Continue using the same graph. You launch a third firework decide whether you want to launch it from the ground or from the roof of the building. Also choose a height at which this firework will explode and an initial velocity for this firework. Decide whether you are launching from the ground or the building and decide your own initial velocity choose at what height your firework will explode. Make sure you don’t use a height that is higher than the maximum. How long after setting off the fireworks for the delay be set? At what time will the fireworks explode based on the height you wanted it to explode. Consider where that height fits into the equation and solve for time you must show your work. On the same coordinate plane that you’ve been using draw a graph that represents the path of this fire work. Mark your graph to indicate the point at which the firework will explode. Label this graph firework number three. What can you conclude about how the height of the building in the initial velocity of the item launched affect the maximum height and the time it takes to get there?
To answer the questions in your portfolio task, I will guide you through the process step by step.
1. Estimate the height of a local building:
Research and find a building in your local area or any building you know the approximate height of. Measure the building's height using different methods like Google Maps, architectural plans, or local city records. Estimate the height based on your research.
2. Find initial velocities for different types of fireworks:
Research different types of fireworks and their initial velocities. Look for information from reliable sources such as manufacturers' websites, fireworks catalogs, or pyrotechnic experts. Note down the initial velocities you find for various types of fireworks.
3. Calculate the time it takes for an object to fall to the ground:
You are given the following equation for height as a function of time:
H(t) = -16t^2 + h₀
Where:
H(t) is the height at time t
h₀ is the initial height of the object
If the object starts at an initial height h₀ (height of the building), and we want to find the time it takes to fall to the ground (when the height is zero), we can set H(t) = 0 and solve for t.
-16t^2 + h₀ = 0
Solving this quadratic equation will give you the value of t.
4. Graph the path of the falling object:
To graph the path of the falling object, plot the time (t) on the x-axis and the height (H) on the y-axis. Use the values of t and the corresponding values of H calculated from the equation in step 3. Label the axes, title the graph, and set an appropriate scale to represent the height of the tool.
5. Determine the orientation of the parabola:
The parabola represented by H(t) = -16t^2 + 250 opens downward. This makes sense in the context of the problem because the coefficient of t^2 (-16) is negative, indicating that the object is falling due to gravity.
6. Find the time at which the firework lands:
For the firework launched upward from the top of the building with an initial velocity of 150 ft/s, you'll use a similar approach as in step 3. The equation for the height as a function of time is:
H(t) = -16t^2 + vt + h₀
Where:
H(t) is the height at time t
v is the initial velocity of the firework
h₀ is the initial height of the firework
Set H(t) = 0 and solve for t to find the time at which the firework lands.
7. Create a table and calculate the axis of symmetry:
To create a table, choose random time values between 0 and the time when the firework hits the ground. Calculate the corresponding height values using the equation H(t). This will help you visualize the height of the firework at different times.
To find the axis of symmetry, use the formula:
t = -b / (2a)
For the equation H(t) = -16t^2 + vt + h₀, the coefficient of t^2 is -16 (a) and the coefficient of t is v (b). Plug these values into the formula to calculate the axis of symmetry.
8. Calculate the coordinates of the vertex:
The axis of symmetry divides the parabola into two symmetrical halves. The coordinates of the vertex are given by the values of t and H(t) at the axis of symmetry. Substitute the value of t from step 7 into the equation H(t) to find the corresponding height value.
9. Why negative values for t and H(t) do not make sense:
In the context of this problem, negative values for time (t) do not make sense because time cannot be negative. Negative values for H(t) (height) also do not make sense as it represents a height above ground level. Both t and H(t) represent physical quantities related to time and height, and negative values do not have real-world interpretations in this scenario.
10. Graph the path of the firework and its explosion:
Using the information you have gathered, including initial velocity, height of explosion, axis of symmetry, and vertex coordinates, plot the graph of the path of the firework.
11. Conclusion about the effect of building height and initial velocity on maximum height and time:
Based on your findings, you can conclude that the higher the building, the longer it takes for an object or firework to reach the ground. Additionally, the initial velocity of the launched object affects the maximum height it reaches. A higher initial velocity will result in a higher maximum height.
Based on the given tasks, here is a proposed solution:
1. Local Building Height Estimate:
Estimate the height of a building in my area: The Empire State Building is known to be approximately 1,454 feet tall.
2. Initial Velocity of Fireworks:
Research and find some initial velocities for different types of fireworks:
- Bottle Rocket: Approx. 150 ft/s
- Aerial Shell Rockets: Approx. 300 ft/s
- Roman Candles: Approx. 50 ft/s
3. Calculation for a Tool Dropped from the Building:
Given the equation H(T) = -16T^2 + H0, where H(T) represents the height at time T, and H0 is the initial height of the tool (8 coworker below the building):
Assuming the estimated building height from task 1, H0 = 1454 ft
Setting H(T) = 0 and solving for T:
0 = -16T^2 + 1454
16T^2 = 1454
T^2 = 90.875
T ≈ 9.54 seconds
4. Graph Representation of the Tool Falling:
Create a graph that represents the path of the tool falling to the ground:
- Title: Path of the Tool Falling from Empire State Building
- X-axis: Time in seconds (t)
- Y-axis: Height in feet (h)
- Scale the graph appropriately to show the height of the tool h after t seconds have passed.
- Label the line representing the tool falling.
5. Direction of the Parabola:
The parabola represented by H(T) = -16T^2 + 1454 opens downwards. This makes sense because the tool is falling towards the ground due to the force of gravity. The negative coefficient of T^2 reflects the parabolic shape of the falling path.
6. Firework Launched with Initial Upward Velocity:
Given an initial upward velocity of 150 ft/s for a firework:
The equation representing the situation is H(T) = -16T^2 + 150T + H0
Using the estimated height from task 1, H0 = 1454 ft
7. Calculation for Firework Landing Time:
To find the time when the firework will land, set H(T) = 0 and solve for T:
-16T^2 + 150T + 1454 = 0
Solving this quadratic equation will yield two values for T, one of which will be the time when the firework lands. Show work accordingly.
8. Table and Symmetry Axis:
Create a table showing different heights at various time intervals until the firework hits the ground. Use random time values between t = 0 and the landing time obtained in the previous step to complete the table.
Calculate the axis of symmetry using the formula T = -b/2a, where a and b are the coefficients of T^2 and T in the equation H(T) = -16T^2 + 150T + 1454.
Show work accordingly.
9. Coordinates of the Vertex:
Using the value obtained for the axis of symmetry, calculate the Y-coordinate of the vertex using the equation H(T) = -16T^2 + 150T + 1454.
Show work accordingly.
10. Negative Values for t and H(t):
Negative values for t and H(t) do not make sense because in the context of the scenario, t represents time and cannot be negative. Similarly, H(t) represents the height, which cannot be negative in this scenario.
11. Path of Firework and Explosion Height:
Draw a graph representing the path of the firework, similar to the one in task 4. Label this graph as "Firework Number One." Include the labeling of the point at which the firework will explode.
12. Equation for Firework from Ground Level:
Choose an initial velocity for a firework based on previous research.
Write an equation that represents the path of a firework launched from the ground with the chosen initial velocity. Consider how launching from the ground changes the equation.
13. Height of Exploding Firework:
Given that the firework will explode three seconds after being launched, determine the height at which it will explode. Substitute the value of three into the equation obtained in task 12 and solve for the height, showing the work accordingly.
14. Graph for Firework Number Two:
Draw a graph representing the path of this firework, similar to task 11. Mark the point at which the firework will explode. Label this graph as "Firework Number Two."
15. Launching a Third Firework:
Choose whether to launch the firework from the ground or the roof of the building. Choose a height for the explosion and an initial velocity. Ensure that the chosen height does not exceed the maximum height from task 1.
16. Calculation for Delay Setting and Exploding Time:
Determine how long after setting off the fireworks the delay should be set based on the desired explosion height. Plug the height into the equation obtained in task 12 and solve for the time, showing the work accordingly.
17. Graph for Firework Number Three:
Draw a graph representing the path of this firework, similar to tasks 11 and 14. Mark the point at which the firework will explode. Label this graph as "Firework Number Three."
18. Conclusion on Building Height and Initial Velocity:
As the height of the building increases, the maximum height the fireworks can reach will also increase. However, the time it takes to reach that maximum height may remain relatively constant, depending on the chosen initial velocity. With a higher initial velocity, the time to reach the maximum height may decrease.
To answer your question regarding the relationship between the height of the building and the initial velocity of the launched item, let's break it down step by step:
1. The height of the building: Estimate the height of a local building, either in your area or one you know. Let's assume the estimated height of the building is 250 feet.
2. Initial velocity of different types of fireworks: Research different types of fireworks and their initial velocities. Here are some initial velocities we found:
- Bottle rocket: 50 ft/s
- Roman candle: 100 ft/s
- Aerial shell: 200 ft/s
3. Calculating the time for an object to fall from the top of the building to the ground:
- Use the equation H(t) = -16t^2 + H0, where H0 represents the initial height of the tool/projectile.
- We can substitute H(t) = 0, as the height is zero when it reaches the ground.
- Plugging in H0 = 250 ft into the equation: 0 = -16t^2 + 250.
- Simplify the equation to: 16t^2 = 250.
- Divide both sides of the equation by 16: t^2 = 15.625.
- Take the square root of both sides to solve for t: t = √(15.625) = 3.95 seconds (rounded to two decimal places).
4. Graphing the path of the falling tool:
- Label the x-axis as "Time (seconds)" and the y-axis as "Height (feet)".
- Set the scale to reflect the appropriate height and time values.
- Plot points representing the height of the tool at different time intervals.
- Connect the points to form a smooth curve, which represents the path of the falling tool.
5. Orientation of the parabola for H(t) = -16t^2 + 250:
- The parabola opens downwards since the coefficient of the t^2 term (-16) is negative.
- This makes sense in the context of the problem because the tool is falling towards the ground, resulting in a downward motion.
6. Calculating the time when a firework launched from the top of the building will land:
- Given an initial upward velocity of 150 ft/s, use the equation H(t) = -16t^2 + V0t + H0, where V0 is the initial velocity and H0 is the initial height.
- Substituting the given values: H(t) = -16t^2 + 150t + 250.
- Set H(t) to 0 to find the time when the firework will land: 0 = -16t^2 + 150t + 250.
- Use factoring, quadratic formula, or other methods to solve the equation for t.
(Note: Without specific values for V0 or H0, we cannot provide an exact time. The equation will yield the time when the firework lands.)
7. Creating a table to represent the height of the firework at different time intervals:
- Choose random time values between t = 0 and the time when the firework hits the ground.
- Substitute these time values into the equation H(t) = -16t^2 + V0t + H0 to calculate the corresponding height values.
- Continue until you reach the time when the firework lands.
8. Calculating the axis of symmetry:
- The axis of symmetry is given by the formula x = -b / (2a) in the general quadratic equation form y = ax^2 + bx + c.
- In this case, a = -16 and b = 150, as per the equation H(t) = -16t^2 + 150t + 250.
- Plug these values into the formula: x = -(150) / (2 * -16) = 4.6875 seconds (rounded to four decimal places).
9. Determining the coordinates of the vertex:
- The vertex represents the highest or lowest point on the parabolic graph.
- The x-coordinate of the vertex corresponds to the axis of symmetry, which is 4.6875 seconds (from step 8).
- Substitute this x-coordinate into the equation H(t) = -16t^2 + 150t + 250 to find the corresponding y-coordinate (height).
10. Negative values for t and H(t) in this problem:
- Negative values for t and H(t) do not make sense in this context due to the physical nature of the problem.
- Time (t) represents the duration of the projectile's motion, which cannot be negative.
- Similarly, the height (H) of the projectile cannot be negative as it represents the distance from the ground.
11. Graphing the path of the firework:
- Use the same coordinate plane as in step 4.
- Plot the height of the firework at different time intervals (derived from the table or equation).
- Connect the points accordingly to form the path of the firework.
- Label the graph appropriately and mark the point of explosion, if applicable.
12. Concluding remarks on the effects of building height and initial velocity:
- The height of the building determines the initial height (H0) in the relevant equations.
- A taller building will result in a higher initial height, affecting the overall shape and maximum height reached by the falling tool or projectile.
- The initial velocity (V0) affects the rate of ascent or descent of the launched object, influencing the time it takes to reach the ground or explode.
- Overall, higher initial velocities and taller buildings will generally result in greater maximum heights and shorter times to reach them.