Lesson Goal:

You will be able to apply quadratic functions to projectile motion.
You will be able to apply quadratic functions to other situations. (OPTIONAL)

In Class Activities

Part 1: Warm - Up

1. Solve: LaTeX: 5x^2-10x-2=0
2. Graph: $LaTeX: f\left(x\right)=2x^2-7x+6$ . Write the x-intercepts, the y-intercept, the vertex, and the domain and range.

Part 2: What is projectile motion?

Imagine: If you throw a paper ball into a trash can from across the room, visualize the path that the paper ball will take.

Connect: How is this situation relevant to what we are learning right now?

Wonder: If you want your paper ball to land in the trash can every time, what are the things you can control?

Part 3: Projectile Motion Equation

Projectile motion models how objects move when thrown against gravity.

Note: We are not in physics, so we don't have to learn where these equations come from. Here is a link if you are interested in learning about that.

Here are the equations you need:

$LaTeX: h_{f\:}\left(t\right)=-16t^2+v_ot+h_o\:OR\:h_{f\:}\left(t\right)=-4.9t^2+v_ot+h_o$

The first equation is relevant when you are working in the English units (feet) and the second equation is relevant when you are working Metric units (meters).

All these letters can be confusing, so here is what they stand for --

$LaTeX: h_{f\:}=\:final\:height$ , (feet or meters)

$LaTeX: v_o=\:initial\:velocity$ (feet/second or meters/second)

$LaTeX: h_o=\:initial\:height$ (feet or meters)

$LaTeX: t\:=time\:$ (seconds)

Desmos

https://student.desmos.com/activitybuilder/student-greeting/670593b5e75166af58f9fc1f

ACTION ITEMS:

1. Watch the videos in this lesson if you need more examples.

2. Write pertinent notes in your notebook/journal.

Main Idea for students who didn't attend class:

Apply Quadratic Functions to Projectile Motion

From Physics we know that motion of any object (in a vacuum, so no air resistance) can be modeled by the equation --

$LaTeX: h_{f\:}\left(t\right)=\frac{1}{2}at^2+v_ot+h_o$

where,
$LaTeX: h_{f\:}$ is the final height of the object,

$LaTeX: a\:$ is the acceleration of the object,

$LaTeX: v_o\:$ is the initial velocity of the object,

$LaTeX: h_o\:$ is the initial height of the object, and

$LaTeX: t\:$ is time.

This equation can be used for modelling path of any object when acting against gravity. Some useful numbers to know are the acceleration due to gravity on Earth is measured to be $LaTeX: 9.81\:\frac{m}{s^2}\:or\:32\:\frac{ft}{s^2}$ . So, this equation essentially reduces to --

$LaTeX: h_{f\:}\left(t\right)=-4.9t+v_ot+h_o\:OR\:h_{f\:}\left(t\right)=-16t+v_ot+h_o\:$

for many problems that we will be solving.

Here is an example of one such problem where you have to find when does an object land on the ground:

Here is another example where you have to find when does an object reach it's max height and what is the maximum height:

OPTIONAL - The following topics are typically covered in Algebra 2A. However, this year these topics are optional.

Besides Projectile motion, quadratic functions can be applied to several other situations.

Here is an example of finding dimensions of a geometric figure given some information:

Here is an example of optimizing area given the perimeter:

Here is an example of maximizing revenue when the demand depends on the price of an object:

When you are finished watching these videos, please finish the corresponding problems and worksheet

Let's do some examples together:

Example 1 –

Suppose a particular trebuchet can launch a 10 kilogram stone with an initial upward speed of 24.5 meters per second. Furthermore, suppose that the stone is 8 meters above the ground when it leaves the sling – that is, its initial height is 8 m.

What is the maximum height achieved by the stone during its flight to the enemy castle?
When does that maximum occur?
How long would it take before the stone comes crashing down to Earth (in case it missed the castle altogether)?

Click here to see the solution.

Do the following examples in your small groups.

Example 2 –A ball is thrown straight up from the top of a 128 foot tall building with an initial speed of 32 feet per second. The height of the ball as a function of time can be modeled by the function h(t) = –16t² + 32t + 128. How long will it take for the ball to hit the ground?

Click here to see the solution.

Example 3:A ball is thrown straight up from the top of a 288 foot tall building with an initial speed of 48 feet per second. The height of the ball as a function of time can be modeled by the function h(t) = –16t² + 48t + 288. When will the ball reach a height of 320 feet?

Click here to see the solution.

Part 4: Reflect on your learning.

-What was the most difficult part of doing examples 2 and 3 for you?

Worksheet pdf