# Predicting the Velocity and Trajectory of a Projectile

## Objective

Students will use their understanding of the horizontal and vertical components of a projectile's velocity to construct a projectile's trajectory for its entire flight.

#### Big Idea

Mathematical models can help solve complex projectile motion problems.

## Overview

The goal of this lesson is to help students learn that models can be extended to solve more complex problems. This lesson addresses the HSA-RE1.A.1 and HSG-SRT.C.8 standards as a way to effectively construct the trajectory of a projectile with known launch angle and initial velocity. It relates to the NGSS Practice of Using Mathematics and Computational Thinking (SP5) because students are asked to correctly choose a solution method for predicting the velocity components along a projectile's trajectory. This lesson also is aligned to the NGSS Cross-Cutting Idea of Systems and Models because students must use both mathematical models and tracking software to leverage their understanding of a projectile's motion to help construct an object's trajectory.

Within this lesson, students begin by using the concept that a projectile has a constant horizontal velocity to add information to a sketch of a sample trajectory. Students then will use image tracking software to construct a parabolic graph of the object's motion.  I ask my students to identify connections between the factors that affect a projectile's trajectory and concepts like trigonometric ratios and Pythagorean theorem they already know. Within this lesson, I discuss the idea that Pythagorean theorem and trigonometric ratios of motion as important models for understanding projectile motion. I ask students to look for and take note of more direct connections between the factors from this week's lessons and the trigonometric identities they study in math class as the lesson progresses as I circulate the room throughout the lesson. I will introduce a new software analysis tool called tracker that students use to track the parabolic trajectory of an Angry Bird. This software tool gives students another way to model the motion of a projectile and helps students overlay velocity vectors onto the parabolic trajectory of an Angry Brid. I choose this pattern recognition tool because I want my students to use tools common to the physics of projectile motion.  I assess student understanding throughout the lesson using informal check-ins, and will assess each student's work at the end of the school day.

## Bell-Ringer

5 minutes

During this part of the lesson I introduce the goals of the lesson and key ideas for students to focus on throughout the lesson.

Today's additional piece of information is a BIG IDEA which states that mathematical models can help solve complex projectile motion problems. Later on within this lesson I ask students to use a simulation to test their predictions of the trajectory of projectiles with known initial velocity and launch angle.

## Looking At A Parabolic Trajectory At Different Time Points

15 minutes

During this section, I project on the interactive whiteboard at the front of the room a parabola of a projectile launched in the forward direction with an initial velocity of 15 m/s at 45 degrees. I ask students to write predictions on worksheet that I have blown up to 11" x 17" about the horizontal velocity at different points of the projectile's path: at the beginning of the launch, at the peak of the trajectory, and near the end of the trajectory. This activity asks students to incorporate information from previous lessons that we will use to understand physics content in this lesson.

In this section of the lesson, I present students with the idea that the horizontal motion of a projectile remains constant.

• First I ask students for the equation for the initial horizontal velocity
• Then I give students the projectile's velocity at t =1 s
• I asks students to recall that the vertical velocity changes by 9.81 m/s every second on Earth
• I ask students to find the vertical velocity at t = 1 s
• I ask students to find the horizontal velocity at t = 1 s using Pythagorean Theorem
• I ask students to compare the horizontal velocity they found at t =0 s and the horizontal velocity at t = 1 s
• Students note that the horizontal velocity is the same at t = 0 s and at t = 1 s

Students do not need Pythagorean Theorem to prove that vx is constant. However, it helps to drive home that vx is constant if students are given the resultant velocity and find vy = viy -9.81 m/s after 1s if students use Pythagorean theorem to check the value of vx and find that vx = vix. Once students have used their mathematical models to identify the initial horizontal velocity, they have determined the horizontal velocity for the projectile while it moves under the influence of gravity. I use a series of guiding question because I am showing students new material that extends their current level of understanding.

After I have introduced this idea using this example problem, I spend the next five minutes of this section calling on students from around the room and write down their ideas on the horizontal component of a projectile's velocity vector. Click here to see a solution to this activity. Students spend the last ten minutes of this portion of the lesson transferring this information into their lab notebooks.

## Investigating Vertical Velocity

45 minutes

During this section of the lesson, there are many activities that students must complete. First students note the characteristics and useful mathematics for understanding projectile motion. Then students must calculate the vertical velocity of a projectile at multiple points on its parabolic trajectory. Next students try out a new software that generates a parabolic trajectory and velocity data for an object it tracks using pattern recognition. Finally, students compare the their calculated values and the velocity data the software generates.

During the next twenty minutes, I project a parabolic trajectory with unknown values on the interactive whiteboard at the front of the room for students to complete. I distribute individual handouts with a projectile's trajectory and ask students to solve for the vertical velocity of a projectile at various instants of time. Students choose their own values for initial velocity and launch angle and apply the concepts from the previous section of this lesson to a similar problem of their own. Click here for an example of student work.

I lead a whole class discussion for five minutes where I summarize the information found here. Students use the Tracker software to generate the parabolic trajectory of a projectile. The program keeps track of an object and generates information about that object’s position, velocity, and acceleration. Students can overlay arrows to represent each vector onto the raw footage. I demonstrate how to select an object to track and choose a window size for the software to search for each frame. Students may choose a projectile motion video from the software library as input for the tracker software.

I remind students of the digital resources available to them which include laptops with the software pre-loaded on them and then ask for resource managers to gather the materials provided at the front of the room where the equipment is kept. After twenty minutes, I circulate while students use the tracker software to test values of velocity students determine using both Pythagorean theorem and trigonometric ratios.

During this section of the lesson students:

• load a video of the first level of Angry Birds
• select a start frame and end frame to analyze
• identify an object for the software to track automatically
• select a window to search within for each frame
• approve or discard any outliers that the software asks user feedback on
• select the variables from a list of choices including(x, y, v, t, etc.) for the software generate kinematics graphs and table for the Angry Bird

Students record both their predictions and the software generated values for vertical velocity at different time points in their lab notebooks.The digital resources include Laptops and the tracker software.

## Closure

10 minutes

I conclude this lesson with a closing question. To close this lesson, I project a drawing An Angry on the interactive whiteboard at the front of the room being launched in the forward direction from the ground at an angle of 30 degrees with an initial velocity of 12 m/s and ask students to construct the trajectory of the Angry Bird. I ask students to complete this task in their lab notebooks and remind them that I will conduct notebook checks at the end of this week as the marking period is wrapping up. Click here to see an example of student work for this section of the lesson.

This type of closure activity closely mimics the new and novel section of the performance based assessment that each student will have to complete in order to graduate.  This section is aligned with the Cross Cutting Concept of Recognizing and Using Patterns Within Systems for Scientific Understanding because students have to use the constraints of a projectile's motion to correctly build its trajectory. I want students to become comfortable with constructing explanations based on real world data.