Example JITT Activity

Summary

Context

This is the second of two lessons on one-dimensional kinematics. The first lesson introduced the relevant physical quantities, position, velocity and acceleration. In this lesson we review the material from the first lesson, further develop the subject, and introduce some applications.

This lesson, including the student responses, is taken from Physics 110, Introductory Physics, at the United States Air Force Academy, Spring Semester 2003. This is a calculus-based course, taken by all the students, regardless of major. The instructor was Gregor Novak.

Assigned Readings

Physics for Scientists and Engineers (3rd Edition) by Wolfson and Pasachoff

Chapter 2: Motion in a Straight Line

Warm Up Questions

1) A car skids to a stop with locked brakes and leaves skid marks 30 m long. What information about the car would the police need to get to be able to estimate how fast the car was traveling when the driver slammed on the brakes?

2) A basketball player jumps 1 meter high off the ground, turns around and starts back down.

Estimate the time she is within 30 cm of the top of her trajectory (her hang time).

(HINT: Calculate the time it takes to fall 30 cm from rest and double it.)

3) It is possible for an object to have, at the same time...

a. zero velocity and zero acceleration?

b. non-zero velocity and zero acceleration?

c. zero velocity and non-zero acceleration?

d. All of the above are possible.

Selected Student Responses

The following are selected for in-class discussion from a set of 32 submissions. (98% of the class responded.)

From responses to question 1.

Ten out of thirty two responses suggested that the information needed was the time. Ten out of thirty two responses suggested that the information needed was the acceleration

In both cases the responses range from very sketchy to fully complete and correct. The sample to take to class consists of three of each.

1_1. The police would need to know how long the vehicle took to come to a halt. Then, they'd have the distance the car moved and the time it took to stop. Velocity is determined by taking distance over time. Thus, the velocity would be found.

1_2. If the officer knew how long the car skidded for (how long witnesses heard the screech) he/she could easily determine the original velocity by using the equation: v=v(knot) + a(t), plugging in -30 m/s/s for the average velocity, t equal to the number of seconds and setting the final velocity to 0.

1_3. If you knew the time it took for the car to stop, you could work backwards to solve for the initial speed by calculating the avg speed (30m/time), then multiplying it by 2 to find the initial speed.

1_4. The rate at which the car would decelerate.

1_5. Since v final^2 = v initial ^2 +2ax, all that is needed to be known is the acceleration of the car as it slowed down.

1_6 .In order to find out the initial velocity of the car (when the driver first slammed on the brakes), the only information that is needed is what the acceleration of the car was as it came to a stop. Since it was stopping the acceleration will be negative, and factors such as the weight of the car, the type of tires, and anything else that contributes to the braking quality of the car. Most likely the manufacturer already has done their own tests and has the average braking acceleration of the specific vehicle, but that doesn't factor in variables such as road condition (was it wet or slippery) or the condition of the tires (if they were mostly worn or had good tread).

From responses to question 2:

Question 2 is a straightforward application of an equation from the previous lesson and most of the students 26 of 32 had no problem with it. Response 2_1 is an example of the conceptual confusion in a small subgroup of the students. We'll use this response to review the

2_1. It takes about 1 second to fall 1 meter. Therefore it takes .3 seconds to fall 30 centimeters. Therefore she spends about .6 seconds within 30 cm of the top of her trajectory.

2_2. 13.In order to complete this problem, I first converted my units so that they would all match. Therefore, 30 cm became 0.3 m since there are 100 cm in 1 m. Next, I decided to convert the time it takes to fall 0.3 m from rest. In order to do this, I used the equation, X = Xo + Vo*t + 0.5*a*t*t. Since I am calculating the time is takes the person to fall from rest, I can assign Vo to the value of 0 m/s. I can also assign x = 0.3m and Xo = 0m. Knowing that a = 9.8 m/s*s, I plug in all of the known values into the equation (0.3m = 0 m + (0 m/s * t) + 0.5 *(9.8 m/s*s)*t*t) and get the answer of t = 0.25 s. In order to get the time the basketball player is within 0.3 m of the top of her trajectory, I must double my answer. Therefore, the final answer comes to t = 0.5 s.

From responses to question 3:

Number of respondents who chose a: 2

Number of respondents who chose b: 3

Number of respondents who chose c: 1

Number of respondents who chose d: 26

From Comments, a response category featured on every assignment:

Comment_1: #3 above seemed kind of tricky. A and B both seemed to make sense and be true, but C was more fuzzy. The only thing I could think of was that the instant something that had zero velocity and zero acceleration started moving, it would have zero velocity, but non-zero acceleration, but I wasn't sure if that was possible. Could you go over that one a little bit in class?

Comment_2: I am a little confused on number 2 with regards to whether or not I should make the acceleration negative since the person is falling and whether the X should equal -0.30 m since they are falling downward. I know that the way I did it I would get the same answer since I know t must be positive, but I am still kind of confused as to when assign negative values for problems such as these.

Comment_3: I think the car problem could also be solved using a kinetic friction constant for rubber on dry pavement, and then using the mass of the car...but I don't know that equation.

Comment_4: This was a fairly easy pre-flight today. The only thing that I didn't think the book explained very well was the derivation of the motion formula that doesn't have time in it.

Comment_5: Took a few minutes to set up a plan of attack for the second problem, but I managed to figure it out easily.

The Class That Follows

The 50 minutes classroom time is divided into three parts: review of the material from the previous lesson (~15 minutes), introducing new material and some applications (~20 minutes), worksheet work in small groups (~15 minutes).

Part 1: The previous lesson introduced the basic definitions of kinematics quantities and the fundamental uniform acceleration equations:

v = v₀+ at

x = x₀+ v₀t +1/2 a t²

v_av= (v₀ + v)/2

The review starts with student response 1_1. "The police would need to know how long the vehicle took to come to a halt. Then, they'd have the distance the car moved and the time it took to stop. Velocity is determined by taking distance over time. Thus, the velocity would be found."

Discuss with the class what is right and what is wrong with this response. Which velocity is determined by the procedure suggested?

Move on to responses 1_2 and 1_3. Let the class decide if these are valid. The discussion allows for a review of the basic definitions and equations.

Let the class discus how realistic is it that the police could actually determine the time.

Move on to the responses to question 2. Walk the class through the basic kinematics equations. In the process let the class deal with response 2_1 in a positive way. There were several responses (not taken to class) that disclosed similar thinking. Try to determine how much of this is genuine misunderstanding and how much, if any, is shooting from the hip by students who simply did not invest enough time.

Then move on the response 2_2. Let the class comment on it. Solicit opinions on how to respond to Comment_2. This issues id probably in the back of many students' mind, even if they don't articulate it.

Part 2: This is the "new" material. Eliminating the time variable between the basics, leading to the fourth kinematics equation: v² – v₀²= 2a(x – x₀).

First ask the class if they agree with response 1_4. Let the class verbalize the solution strategy, i.e. we need an equation linking which kinematic variables? How is that related to Comment_3?

Show Comment_4 and ask: how many in the class agree with that.

Move on to the derivation of: v²– v₀² = 2a(x – x₀). Show responses 1_5 and 1_6.

Finally, time permitting, let the class discuss question 3, which actually had a good response rate. Show Comment_1. It is quite likely that many correct responses were not underpinned by real understanding that students could articulate.

Part 3: Divide the class into groups of two or three and bring out the worksheet. Let the groups respond to the MC questions first. Use clickers if the class has a Personal Respond System in place.

Finish the class with as many worksheet problems as possible. It is virtually impossible to complete all the parts of the worksheet.