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**INTRODUCTION**

In an introductory physics class, students need to understand the relationship between the motion of objects and the forces they experience. Newton’s first law of motion helps students predict the qualitative motion of an object given the known forces exerted on the object. For example, if the sum of all the forces or the net force on an object is zero, the object will have a constant velocity. If the object is at rest, it will remain at rest; and if the object is moving, it will continue moving at constant speed in a straight line.

This can also go the other way. The observed motion of an object can be used to make an inference about the forces experienced by the object. For example, if an object is moving in a straight line and decreasing speed, it can be inferred that the sum of all the forces exerted on the object must be pointing in the opposite direction of its motion.

In an introductory physics class, students will also need to have a quantitative understanding of the relationship between the motion of objects and the forces they experience. This comes in the form of Newton’s second law of motion. Rather than directly giving your students this equation, give your students an experience where they can develop this equation through a guided investigation.

The following article describes an introductory physics lab that allows students to develop a model of the relationship between an object’s acceleration and its mass. The algebraic equation representing this relationship will be Newton’s 2nd law of motion.

The following equipment is needed for **each** lab group.

- (1) Fan Cart
- (1) Slotted Mass Set with hanger
- (1) Car and Ramp Lab
- (1) Physics Workshop Stand
- (1) Digital Newton Meter
- (1) Whiteboard

This lab incorporates the modeling method of instruction, which involves developing a procedure, data collection, data analysis, and collaborative discussion. My descriptions of the lab are based on my classroom experience with students and outline one of many ways this can be done. The bigger picture I want you to see is how you can guide your students to play an active role in each part of the investigation.

One video can not paint a complete picture of what this looks like in the classroom, but it can give you a taste. If you are interested in learning more about these types of guided inquiry labs used in the modeling method of instruction visit www.modelinginstruction.org.

**PRE-LAB DISCUSSION**

During the pre-lab discussion, you want to present the students with a scenario that helps them link the concepts of acceleration and mass, and get them thinking about how the two variables are related.

Start by demonstrating the motion of an object that increases speed when released from rest like a fan cart. Ask students why the fan cart’s speed is increasing once released. Based on their understanding of Newton’s 1st law of motion, the students should be able to explain that the fan cart is accelerating because the sum of the forces is not zero. The sum of the forces must be pointing in the direction of the cart’s acceleration.

Ask students how the cart’s acceleration could be measured or calculated. At this point in an introductory physics class, students should be familiar with measuring or calculating an object’s acceleration. Students could measure the cart’s acceleration using a motion detector, or students could calculate the cart’s acceleration with collected time and distance measurements. Tell students that scientists and engineers often want to go beyond accurately describing what happens, to accurately predicting what will happen. Tell students that they are going to a lab investigation which develops a way to predict the size of the cart’s acceleration.

Before you guide students to develop a procedure for this lab, the students will need to come up with variables they think will affect the cart’s acceleration. The students will then investigate how one of these variables is related to the cart’s acceleration. Start by asking students what things they could change about this situation that affect the cart’s acceleration. The students will list things such as the cart’s mass, the size of the force of the air pushing forward on the propellers, and the size of the combined frictional forces on the cart. Help guide students to see that both the force from the air and the frictional forces both affect the size of the sum of the forces on the cart, and it is the size of the sum of the forces that affect the cart’s acceleration.

There are now two experiments that could be performed. Either the students could find the relationship between the cart’s acceleration and the sum of the forces, or the students could find the relationship between the cart’s acceleration and its mass.

In order to get a wide range of at least six different values for either experiment, the students would need to use either six different fan cart masses or have the fan experience six different size forces from the air. This fan cart from Arbor Scientific does allow you to vary the fan’s force by changing the fan’s speed with the use of an included metal spacer. However, only two different speeds are possible.

Since the fan cart’s mass is easily changed with the addition of slotted masses, it makes more sense for the students to investigate the relationship between the fan cart’s acceleration and its mass. Write the following purpose on the board: to determine the relationship between an object’s acceleration and its mass.

**DATA COLLECTION**

Now you can help guide students through a procedure for collecting the needed data. To determine the relationship between an object’s acceleration and its mass, the students will need to collect a variety of different acceleration and mass measurements. During the pre-lab discussion, the students already discussed different ways to measure the cart’s acceleration. Each lab group will just need to decide which of the discussed methods to use. Students could either measure the cart’s acceleration using a motion detector or students could calculate the cart’s acceleration with collected time and distance measurements.

During data collection, the students will need to change the fan cart’s mass and measure or calculate its acceleration. Remind students the sum of the forces on the fan cart should remain constant throughout the entire lab, so any changes in the acceleration are a result only of a changing mass.

If you have motion tracks with height adjustments, discuss how a small decline can be used to minimize the effect of the frictional forces on the fan cart. Students should adjust the incline’s angle until the cart moves at a constant speed after a small push. If the cart’s speed is relatively constant, the component of gravity parallel to the incline will make up for the frictional forces experienced by the cart. If this is done, then the sum of the forces on the fan cart will be equal to the force on the fan blades by the air.

* Note:* When using the metal spacer to make the fan spin slower, the force is relatively small, and it makes data collection challenging with larger masses. In order to collect samples with larger masses, angle the ramp to minimize friction, give the cart a little push to get started, and use a motion detector to find the cart’s acceleration using the slope of a velocity versus time graph.

In order to introduce some variability in the collected data, give half of the lab groups fan carts with two batteries and give the other lab groups fan carts with one battery and one spacer. You can either do this without the students’ knowledge or be transparent about it. This variability in the data will help the students reach conclusions about their collected data.

**DATA ANALYSIS**

To analyze the collected acceleration and mass data, have the students make a graph by placing the acceleration values on the y-axis and the mass values on the x-axis. Students should find that the graphed data will produce a hyperbolic trend. In my class, this is the first time students have encountered a nonlinear graph when analyzing lab data. Writing an algebraic equation for a nonlinear relationship is a skill I will introduce at this time.

Most graphing software has the option to add different types of curve fits to the graph, but I want my students to re-express their data so the final graph appears linear. This is sometimes called “linearizing” a graph. If the graph appears linear, the students can simply add a linear fit and write the corresponding linear equation as they have done before. If you’ve never heard of re-expressing or linearizing a graph, the following is a brief introduction. The shape of a graph can be changed by simply modifying the values on either the x or y-axis. Below are several graphical trends commonly found in nature. The middle row shows the ways in which these common graphical trends can be linearized. A linear fit of the re-expressed or linearized graph can be used to write the algebraic equation showing how the two variables are related.

When students graph the fan cart’s acceleration as a function of mass, the graph will appear hyperbolic. There are two common ways to linearize a hyperbolic graph, so students will potentially need to try both methods. To linearize a hyperbolic graph, the variable graphed on the x-axis either needs to be inverted or inverted and squared. To make a graph of acceleration versus the inverse of mass, a new column of inverse mass values will need to be calculated. The students would then graph the acceleration values on the y-axis and inverse mass values on the x-axis. When making a linearized graph, whatever manipulation is done to the variable must also be done to the units. The units of inverse mass would then be one divided by kilograms. This graph should appear linear. Adding a linear fit allows the equation to be written that shows the relationship between the cart's acceleration and mass.

If students first try to linearize their original graph by graphing the acceleration versus the inverse of mass squared, they will get a side-opening parabola. Encourage students to continue trying different ways to linearize or re-express their graph until they get the straightest possible trend. Depending on the quality and range of the data, it is sometimes difficult for students to determine which graph is most linear. Encourage students to use the linear correlation value, or the square of this value, as an objective way to measure the amount of linear association between the graphed variables. The final linearized graph will be the one where the absolute value of the correlation or the R squared value is closest to 1.

Teaching students how to linearize or re-express their nonlinear graphs might seem like a lot of extra work, but the payoff will come during the conclusion discussion. Before the students circle up to share the analysis of their results, ask each lab group to discuss the shape of their graph and the significance or meaning of both the slope and the y-intercept of their linearized graph and equation.

**CONCLUSION DISCUSSION**

To facilitate a whole-class conversation about the relationship between an object’s acceleration and its mass, have each lab group record their original graph, linearized graph, and resulting equation on a large whiteboard. Below is an example of what one lab group’s whiteboard would look like.

Have the class circle up so that everyone can clearly see the graphs and equations on each whiteboard. Remember that your goal is to help facilitate a conversation that allows your students to make connections and draw conclusions from the graphs and equations

Start by asking students to compare the graphs and equations on the whiteboards and identify any similarities or differences they see. Since different numbers of batteries were used in the fan carts, the students will be able to identify similar graphical shapes and y-intercept values, but different groupings of slope values. The lab groups who used fan carts with two batteries will have higher slope values than the groups who used fan carts with one battery and one spacer. This will be helpful when students discuss the meaning of the slope. Once the similarities and differences are identified, the rest of the conclusion discussion should focus on the meaning of the slope and the significance of the y-intercept from the linearized graph and equation.

When students look at the values of the y-intercepts, they will have small positive or negative values. The question becomes whether these values are significant or insignificant. To help students judge the significance of a y-intercept, it is not enough to just look at the value. The value of the y-intercept must be compared with the range of values collected on the y-axis, in this case, the acceleration values. The threshold is somewhat arbitrary, but I tell my students that the y-intercept is insignificant if it is less than 5% of the maximum y-value.

The y-intercept of the linearized graph can also be considered insignificant if it can be reasoned away. Have the students think about what they would expect the acceleration to approach if the value of the inverse mass approached zero. It helps to have the students first identify how the mass value would have to change to get the inverse mass value to approach zero. Through a guided conversation, students will be able to identify that the inverse mass value will approach zero as the mass gets larger and larger. As the fan cart's mass approaches a very large value, the acceleration would be expected to approach zero. Since the y-intercept can be reasoned away, it can be left out of the algebraic equation.

When students look at the slope of the linearized graphs and equations, they should find two groupings of values. Ask students why there are two groupings of slope values. If students can determine the cause of different slope values, this will help them determine the meaning of the slope. The slope of any linear relationship represents the change in the y-variable for each unit of change in the x-variable. For this lab that would be the increase in the cart’s acceleration for 1 unit of increase in inverse mass. This may be a true statement, but it’s not helpful. It turns out that the slope represents the size of the sum of the forces on the fan cart, but how do you guide students to that conclusion?

Start by reminding students that the slopes of their linearized graphs are constant. This is true for any linear relationship. So this means that the slope must represent something that stayed constant during each trial in their experiment. This constant thing must also explain why groups got different slope values. Discuss each shared idea to help students decide if this is the meaning of the slope. Remember, the slope value is equal to the sum of the forces on each group’s fan cart, but you want students to identify this.

Here’s another way to help guide students toward the meaning of the slope. Point out two whiteboards, one with a significantly higher slope than the other. If both of these groups used the same mass, the group’s fan cart with the higher slope value must have experienced a larger acceleration. Why would two objects with the same mass experience different accelerations? Based on the prelab discussion, students should recognize that two objects with the same mass would experience different accelerations if the sum of the forces was different on each. Ask students which lab groups had fan carts that were pushed with more force. Students will realize that lab groups with higher slope values used fan carts that were pushed with more force.

Making this correlation does not prove that the slope is equal to the sum of the forces. More evidence is needed to reach that conclusion. Suggest that the size of the sum of the forces on each fan cart could be measured with a force sensor or “digital newton meter”. Remember that the sum of the forces on the fan cart is equal to the size of the force on the fan cart by the air. Measure the approximate size of the sum of the forces on a fan cart with two batteries and the size of the sum of the forces on a fan cart with one battery and one spacer. These force measurements should be very close to the slope values. This shows that the slope in each group’s equation does represent the size of the sum of the forces on the fan cart.

After the students reach a consensus about the meaning of the slope and the significance of the y-intercept, you can finally write the general equation on the board. Tell students this equation is known as Newton’s 2nd law of motion, which shows the relationship between an object’s acceleration, mass, and the sum of the forces.

If you are interested in learning more about these types of guided inquiry labs, used in the modeling method of instruction, visit www.modelinginstruction.org.

Happy investigating!

Aaron Debbink

**Aaron Debbink**

Physics Instructor

Indian Hill High School

Cincinnati, OH

Aaron Debbink is a physics teacher with 16 years of classroom experience who has an undergraduate engineering degree and a masters degree in physics. He is passionate about building a classroom culture that values exploration and a drive for understanding. Aaron uses Modeling Instruction in his introductory and AP physics classes and has been a Modeling Instruction workshop leader since 2011. Aaron is a Knowles Teaching Fellow and the recipient of a 2019 Yale Educator Award.