The Bootstrap:Physics materials are the result of the Computational Modeling in Physics First with Bootstrap project, a collaboration between the AAPT, AMTA, Bootstrap, and STEMteachersNYC. We especially want to thank Melissa Girmscheid, Jess Dykes, Bryan Battaglia, Holly McTernan, John Baunach for their contributions to the project.

The "Physics First"-based course consists of five units:

## Qualitative Energy

Students begin by defining a system and studying the conditions present in the system (the state of the system), identifying the changes in the state of the system between two moments in time, and then identifying the mechanisms by which these changes occur. This pedagogical progression results in the development of an operational definition of energy - a quantifiable measurement of the ability of a system to change - and a knowledge of energy storage modes, transfer mechanisms, and, in some systems, conservation.

Key representational tools (e.g., system schemas, state diagrams, energy bar graphs, and computational simulations) are developed throughout the unit. By the end of Unit 1, students are able to describe a number of complex systems and are prepared to dive deeper into many of the phenomena introduced - specifically those involving motion.

## Constant Velocity

This unit introduces students to using laboratory tools to make quantitative measurements, specifically of position and time, and many new representations available to them when making quantitative observations (e.g., data tables, motion maps, computer simulations, position vs. time graphs, and velocity vs. time graphs).

The unit begins with a paradigm lab, collecting measurements of the changing position of an object with relatively constant motion at various clock readings. Using this collected data, students create a motion map and a computational simulation of the object, resulting in the development of the relationship between the position at one moment in time and the next. Critically, students first discover the pattern of the object’s next position at some future time being the current position plus the change in position during that time interval, and later revised to the current position plus the product of the velocity and the time interval. This progression leads to the operational definition of velocity as the ratio of the change in position over the change in time.

## Uniform Acceleration

1) Position can now change at a non-constant rate. Whereas Unit 2 investigated constant velocity motion, Unit 3 investigates changing motion - specifically, motion with uniform (constant) acceleration. Position in the horizontal direction will continue to be identified by the variable x and vertical position will be identified as y (horizontal and vertical displacement will also be represented by Δx and Δy, respectively).

2) It is important to maintain the distinction between speed and velocity made in Unit 2; speaking of a change in velocity is more detailed than simply discussing a change in speed, as velocity makes reference to the speed and direction of the object.

3) The differential function representation introduced in Unit 2 is seen again in Unit 3. Velocity, constant in Unit 2, will now change iteratively by some constant amount Δv over some time interval Δt. This structure echoes the differential function of the position function in Unit 2 and follows naturally as a consistent way to think about change. This iterative, differential approach stays as the philosophical basis of students’ computational representation of motion.

4) Motion maps remain semi-quantitative devices that represent the object’s position at evenly- spaced clock readings. As in Unit 2, this representation continues to drive the development of the computational representation of motion. The space between dots grows larger as an object moves faster, and shrinks as an object slows down; simultaneously, the velocity vectors indicate the (relative) change, by becoming longer or shorter as the speed increases or decreases, respectively. Emphasize to students that both the dot spacing, and the velocity vector length must change consistently to properly illustrate the motion.

5) Acceleration vectors will be introduced to the motion map representation in this unit. Objects with increasing speed will be represented by the acceleration arrows pointing in the same direction as the velocity vectors at each clock reading and have a matching increase in dot spacing and velocity vector size. Objects with decreasing speed will be represented by the acceleration arrows pointing in the opposite direction of the velocity vectors at each clock reading and have a matching decrease in dot spacing and velocity vector size. If an object has constant velocity, no acceleration vectors are drawn, and the motion map looks exactly as it did in Unit 2.

6) The physical significance of the slope of a velocity-time graph is the rate of change of velocity with respect to time. Investigating the relationship between Δv and Δt leads to the definition of average acceleration:

$$\displaystyle \require{physics}\va*{\bar{a}} = \frac{\Delta \va*{v}}{\Delta 𝑡}$$

Teachers may choose to have students simply call this acceleration, since there is no non- constant acceleration analyzed in this unit (mirroring the use of velocity in Unit 2).

7) Acceleration can be either in the positive or negative direction, and the direction (or sign convention given to the acceleration) is not in and of itself an indication of whether the object is speeding up or slowing down. We will not be allowing our students to use the term deceleration.

An extremely common student preconception of acceleration is that a positive acceleration always indicates an increase in an object’s speed, and a negative acceleration always indicates a decrease in an object’s speed. This preconception must be brought forward and dealt with explicitly throughout the unit. A positive acceleration would result in a faster speed if and only if the velocity were also in the positive direction; but a positive acceleration would create a slower speed if the velocity were in the negative direction (akin to a car rolling backwards and the driver stepping on the gas, the change in velocity is positive, and the negative velocity is getting closer to zero before becoming positive). The opposite is also true: A negative acceleration would result in a faster velocity if and only if the velocity were also negative; but a negative acceleration would create a slower speed if the velocity were in the positive direction (akin to a car rolling forward and the driver stepping on the brake pedal).

Acceleration is a much harder concept for students than velocity; while velocity is a rate of change, acceleration is the rate of change of a rate of change. Describe the units of acceleration as meters 'for every' second 'for every' second (rather than the traditional, much more confusing, ‘meters per second squared’). This phrasing highlights the “rate of change of a rate of change” relationship. When writing the units as a fraction, the numerator is the change in the velocity (meters 'for every' second) and the denominator is the change in the time (seconds). Stated as a sentence, an acceleration of 4 m/s/s means that for every 1 second of change in time, the velocity changes by 4 m/s.

8) When interpreting kinematic graphs, continue focusing on the slope of the graph as the rate of change in the physical quantity represented on the vertical axis with respect to the one represented on the horizontal axis. Now we have multiple graphs to consider (the position-time graph, the velocity-time graph, and, to a lesser extent, the acceleration-time graph), and they look very different while still describing the same motion. Students should be able to describe how each graph illustrates that an object is speeding up, slowing down, moving with a constant speed, and in which direction it is moving. They need a thorough grasp of the relationship between the slope of a line on a position-time graph and the velocity. Students should also have a thorough grasp between the slope of a line on a velocity-time graph and the acceleration. Be sure to reinforce the connections between $$\displaystyle \require{physics}\va*{x}$$ vs t graphs and $$\displaystyle \require{physics}\va*{v}$$ vs t graphs. "Stacking" the curves by placing the $$\displaystyle \require{physics}\va*{v}$$ vs t graph directly underneath the $$\displaystyle \require{physics}\va*{x}$$ vs t graph will further illustrate this relationship. Be sure to use a wide variety of graphs, of various difficulty levels. For example, when interpreting velocity-time graphs, begin by focusing on graphs whose line does not cross the horizontal axis (thereby changing direction), before showing graphs of more complex motions, like those for an object rolling up and then back down the ramp. Students should see that the acceleration on such a graph has a uniform acceleration for the entire motion, including the moment that the velocity graph crosses over the t-axis (the moment the object has 0 velocity), the acceleration is never zero during such a situation.

9) The development of kinematic equations is not necessary - students have enough tools to solve all kinematic problems in units 2 and 3 without algebraic representations - but if they are to be introduced, they should be derived from the velocity vs. time graph.

Reinforce that the area under a $$\displaystyle \require{physics}\va*{v}$$ vs t line still represents the displacement of the object between two clock readings, as we did in Unit 2. Emphasize the fact that the area is no longer a rectangle but is now a more complex shape (trapezoid), or a combination of simple shapes (rectangle and triangle).

## Balanced Forces

In this unit, students are introduced to the first half of Newton’s Modeling Cycle:

a) From motions (read: changes in velocity), infer forces
b) From forces, deduce changes in motion

We are moving from the realm of descriptive models (kinematics) to that of causal models: dynamics.

It is essential for students to recognize that a system with constant velocity differs from one with non-constant velocity, and that only changes in velocity require an interaction between an agent and an object. We define this interaction as the concept of force. After the broom ball pre-exploration, students will get the sense that force is required to change the motion of an object. This is reinforced by Activities 2 and 3, and one can use worksheets 2 and 3 as an opportunity to apply the force concept in a qualitative way. It is important to carefully treat how to go about drawing force diagrams in which one represents the object as a point particle. In this unit, we will introduce the concept of the identification of systems and will do so by drawing dotted lines around the system being studied to help students distinguish between the object/system and the agent(s) that affects its motion. Significant attention will also be paid to identifying forces based on the type of interaction, the system, and agents involved in the interaction. This care in building the idea of a force as an interaction will pay large dividends when Newton’s Third Law is introduced in Activity 4.

Some students may notice the connection between the magnitude of the acceleration of a freely falling object (end of unit 3) and the gravitational field strength. Postpone discussion of this connection until Unit 5 (Unbalanced Forces) in which we will quantify the relationship between force, mass, and acceleration. This way students are more likely to understand the g in the Fg = mg equation as the gravitational field strength (desirable) as opposed to the quite different concept of the acceleration due to gravity, whose magnitude just happens to be the same.

Students will practice drawing simple force diagrams which require no vector components, and using those diagrams, the equation for equilibrium ($$\displaystyle \require{physics} \Sigma\va*{F} = 0$$), and F𝑔 = 𝑚𝑔 to write the equations that will allow them to solve for unknown forces.

This unit will also introduce a method for determining the expressions for both static and kinetic friction. We will determine that friction is a function of the force between the surface and the object moving across it, but not the area of the contact, and that there are dramatic differences between the static case and the kinetic case.

When students have gained confidence representing forces and their effects on system motion without the use of vector components then further mathematical treatments can be considered. In this introductory course, we choose not to decompose force vectors using trigonometry. However, it is important to expose students to qualitative analyses of such problems, as not all forces act parallel to the coordinate axes. Additional treatments are offered as supplemental resources for this unit.

In this unit, we build on the understanding of Newton’s 1st Law developed in Unit 4 to include the motion of an object experiencing unbalanced forces. In Unit 4, we had developed the idea that constant velocity is the result of balanced forces (Σ F = 0), and *non*-constant velocity is the result of *un*-balanced forces ($$\displaystyle \require{physics} \Sigma\va*{F} ≠ 0$$). In this unit we will develop a more robust expression, that can explain both the balanced and the unbalanced force situation for both constant and non-constant acceleration, culminating in the commonly used equation $$\displaystyle \require{physics} \Sigma\va*{F} = 𝑚\va*{a}$$ to model Newton’s 2nd Law.