Experience First:

This lesson marks the third and final time that your students will be interacting with particle motion but this time it’s in the context of parametric functions and vectors! Because a foundation has already been established about the relationship between velocity, acceleration, and speed (among others), the focus will be to build a bridge between functions that are rectangular with those that are parametric or vector-valued in nature. The first task students will explore is revisiting the concept of speed as an absolute value and what that actually means (question 1). Try to elicit responses that demonstrate an understanding that absolute value is the distance from the point of origin, whether on a number line or on a coordinate plane, not just the positive version of a value. Question 2 will lead the students to transfer this knowledge to a 2-dimensional vector on a coordinate plane. Once it is clear that on a number line, absolute value is a one-dimensional length, the “aha!” moment will follow when your students realize that absolute value on a coordinate plane is the distance to the origin as found by the Pythagorean Theorem. Most importantly, both represent speed! Pause the class after question 2 to consolidate these ideas and formalize the formula for speed of a vector valued function.

 

The next connection that students will make is with the idea of total distance traveled. With rectangular functions, students have learned to integrate |v(t)| to obtain this result. Question 3 asks students to use their velocity function to find the change in position as they have done with rectangular functions. Note that students are calculating horizontal and vertical displacement separately and students have plenty of familiarity with calculating one-dimensional motion. Next, question 4 presents the idea of total distance and how this might be calculated. If you find students spending too much time on completing the table, you can encourage groups to divide and conquer to find the values. The big takeaway is that total distance is not found by taking the absolute value of the displacement, but by accumulating all the distances traveled over small intervals. Even motion in the negative direction adds to the total distance. Students first find total distance using an approximation and then should realize that integrating the speed function will give the exact total distance. Finally, help them to recognize that in both rectangular and parametric functions, integrating the absolute value of velocity (which represents speed) results in total distance traveled. 

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Monitoring Questions:

• What does absolute value mean?

• How is absolute value related to speed?

• How can you find the distance from a point to the origin?

• Why isn’t speed a vector?

• How can the components of a velocity vector be used to find the change in position?

• What does integrating horizontal velocity represent? What does integrating vertical velocity represent?

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Formalize Later:

To solidify these concepts for students, relate each new topic back to a familiar one with rectangular functions. For example, “remember how we found acceleration when we had a function v(t)” will lead to the realization that taking the derivative of velocity yields acceleration regardless of the type of function given. In a similar way, the concept of speed follows from the rectangular version if students understand what absolute value actually represents (distance from a point of origin). Finally, a comparison of total distance in rectangular vs. parametric functions will make the parallels between the two obvious.

 

Students tend to struggle with remembering the parametric speed expression when it is not connected to a solid reason for its existence. We hope that this exploration will provide the cement necessary to make it stick!

 

Finally, many students struggle with questions where slope is requested when given parametric or vector-valued functions. We included one of these questions in the Check Your Understanding to provide the conversation starter about when dy/dt and dx/dt should be combined into a fraction to represent slope and when they should stand alone.

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Exam Tips:

Particle motion problems can cause issues for students who do not clearly communicate their answers. For example, if a question asks them to find the velocity at t=2 but the answer is written as v(t)=__  instead of v(2)=___, the response may not receive full credit due to the fact that v(t) does NOT always equal that value, just at t=2. This type of error is called a “linkage error” and occurs when students say two things are equal that are not, or are not always, equal. Although students may see this as a “picky point”, remind them that communication is one of the mathematical practices and will be necessary for all written responses.

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Additionally, parametric (or vector) particle motion problems are often the subject of one of the FRQs on the AP Calculus BC exam. Providing lots of practice with these is a great way to get your students ready!