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.