# Projectile Motion: Challenge Problems

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## Objective

Students will utilize their understanding of trigonometry and simulations to analyze a real-world application of projectile motion.

#### Big Idea

A projectile’s motion is governed only by its interaction with the force of gravity.

## Overview

The goal of this lesson is for students to use their understanding of trigonometric identities and the equations of motion to correctly answer problems related to a projectile's motion. This lesson addresses the HS-PS2-1 standard because it asks students solve problems where the net force is equivalent to the gravitational force. It also aligns to the and HSG-SRT.C.6  standard because students must leverage their understanding of trigonometric ratios to solve problems that relate to the horizontal and vertical components of a projectile’s velocity vector. The gravitational force is given by the product of mass and the acceleration due to gravity (also known as the gravitational field strength). It aligns with the NGSS Practices of Using Mathematical Reasoning (SP5) and Constructing Explanations (SP6) for Science because students will use mathematical logic to create both solutions of a projectile’s motion on different planets. This lesson also is aligned to the NGSS Cross Cutting Idea of Patterns because students must recognize that the gravitational field strength which affects a projectile’s motion directly relates to the size of the planet.

Within this lesson, students construct an explanation of the physics that describes a projectile's motion to meet the constraints of their problems. Each problem is on colored card stock. The yellow problems ask students to predict whether the red angry bird will hit its target given a set of initial velocities and launch angles. The blue problems ask students to predict the time of flight of a red bird on different celestial bodies. The red cards ask students to predict the maximum height of an angry bird on different celestial bodies given a set of initial velocities and launch angles. Students then conduct background research to determine factors like the gravitational field strength of different planets and use this background information, along with Galileo’s equations of motion from their notes, to choose a mathematical relationship to help solve the problem. Finally, students work in teams of two to communicate their solution connecting trigonometry and equations of motion to a projectile’s trajectory, time of flight, and range. Within this lesson, I ask students to focus on stretching their prior trigonometry knowledge to address what happens when the Angry Bird launch occurs on a different celestial body. I assess student understanding throughout the lesson using informal check-ins and assess each student's work at the end of the school day.

## Bell-Ringer

5 minutes

This part of the lesson begins with this writing routine where students write the objective and an additional piece of information in their notebooks as soon as they enter the classroom. Today's additional piece of information is a Big Idea which states that a projectile’s motion is governed only by its interaction with the force of gravity.

The objective of the Bellringer is to give students a clear understanding of the focus of today's lesson. In this lesson I want students to get ready to leverage information gathered from their understanding of trigonometric functions to construct mathematical models for the behavior of a projectile on a different planet or under a novel set of constraints.

## Conducting Background Research

30 minutes

I follow the bell-ringer with an overview of finding credible sources. The “Is it credible?” routine consists of:

• Look for a well-known publisher or journal
• Look for the number of times other academic cite the author’s work
• Go to easybib.org
• Paste the URL or ISBN number into the search stream
• Hit the “Cite This” Button
• Check  for a green credible label
• Add any missing information to the citation form
• Select create citation to make an MLA Citation

While students write down the steps to this routine in their notebooks, I circulate and take attendance. Students use this routine to conduct background research for the challenge problem activity. Students may cite, journal articles, our openStax digital textbook, websites like the physics classroom and Georgia State University's "Hyperphysics".

Students are given a choice of challenge problem which are printed on yellow (page 1), blue (pages 2-3) and red (pages 4-5) cardstock. Students read each choice and make a decision based on their current level of understanding. I do not tell students what the colors mean, I ask them to choose between the three options. Students are tasked with creating a coherent and mathematically supported visual that communicates their understanding of projectile motion using information from multiple sources.  Students are expected with working in small teams to conduct background research which they share via google docs. Students use easybib.com and SEO tools plagiarism checker to check the validity and authenticity of their sources. The minimum components of each visual are given below:

1.  Backstory: Give the audience the backstory (what’s the significance of the problem).
2.  Physics of the Problem (P.o.P.):  Give a Quick overview of the problem.
3.  Solution: Step by Step solution including a description of each step, with equations, units and an actual solution with explanation.
4.  Framework: Explain the connections between the physical concepts in your solution.
5.  Tips: Give tips on how to solve similar problems in the future.

A Choice Board on the methods of communication is provided in this section's resources. The choices include:

• A diary entry
• A letter to another physics student
• A blog

During this section of the lesson, students work in pairs to gather information about their back-story and physics of their problem. Students write pertinent information in the notebooks. This is the part of the lesson where students make connections between the problem and projectile motion using mathematical reasoning. Click here to see an example of student work.

## Creating A Visual Representation of Challenge Problems

35 minutes

The goal of this section of the lesson is for students to create a visual that demonstrates their understanding of projectile motion regardless of the planet the motion occurs on. The purpose of this assignment is to show students that step-by-step mathematical reasoning can help simplify challenging problems into more manageable parts. By breaking the problem into parts, a solution appears more viable than the original problem may have seemed at first glance. The use of mathematical logic to simplify solutions and recognizing patterns that relate to projectile motion data is important for several of the lessons within this unit. I also distribute Chromebooks for students who choose to create a visual digitally using Popplet, Prezi, Powerpoint or Powtoon. Click here to an example of student work.

Students spend 2-5 minutes choosing between the visual options. Then students use about 25 minutes to leverage the information in their written explanation as a way to create a visual that meets the criteria from the previous section. The challenge problem choices are on colored card stock. The red problems ask students to predict whether the red angry bird will hit its target given a set of initial velocities and launch angles. The blue problems ask students to predict the time of flight of a red bird on different celestial bodies. The yellow cards ask students to predict the maximum height of an angry bird on different celestial bodies given a set of initial velocities and launch angles. Students then conduct background research to determine factors like the gravitational field strength of different planets.

Students may work on the visual as individuals or in pairs.  As students are creating their visuals I walk around checking-in with students to ensure they have met all of the requirements I describe in the previous section of the lesson. It takes a little longer for some students to get started than others, but after five minutes or so most students decide which visual they are going to create and quickly get to the task at hand. At the end of this section, I pause and ask students to share their solutions with me by sending me the link via email or posting the solution to the Edmodo wall.

## Closure

10 minutes

To close this lesson, I ask students to talk to their tablemates about the different challenge problems that they worked on. Then I ask students to describe one thing they felt confident about during the lesson. I also ask them to describe in their notebooks one puzzle they still had about the topics within the lesson. Student responses on topics they are confident about include, "determining the time of flight" and calculating "the components of the projectile's velocity". Some students had a puzzle about the acceleration due to gravity on other planets and which range equation to use during their calculations. I remind students that I will accept submissions until midnight on Wednesday for our Monday lesson.