Can terminal velocity be determined from the information given?

[feynmanshomeboy]

Homework Statement

From Serway, 10th edition 2.81

If acceleration of MINI cooper S is modeled as a = a0 - kv, where a0= 4.0 m/s^2 . . .

What is the car's max speed and how long does it take to reach 95% of max speed?

See full screenshot of problem and my solution in attached pdf.

Thank you in advance for your assistance so I can pass on the knowledge to my students.

Relevant Equations

Linear drag of the form: a = a0-kv

I suspect that I have the entire explanation and solution, but I am wondering:

Is it possible to be able to find the maximum velocity of the car from the information given?

The end of the problem says, "At max. accel., how long does it take the car to reach 95% of its top speed?" which seems to imply that a numerical answer is possible without expressing the answer in terms of an unknown max speed or just looking up the car's max speed -- available from the car maker.

The problem and my solution are in the attached .pdf.

I referenced Classical Mechanics by Taylor and noted that in all cases I saw, terminal velocity or the "characteristic time" parameter were determined by the viscosity modeling using mass and fluid density which is not at play here. In the general case, Taylor focuses percentage of terminal velocity corresponding to a multiple of of the "characteristic time."

I am a high school teacher and I want to ensure that I have all possible insights into the problem.

 

[Frabjous]

There are two independent constants in the acceleration equation. Therefore you need two pieces of information to determine them.
In regards to the .95, think about how long it will take to reach maximum velocity.

 

[kuruman]

Where did the maximum speed of 150 mph in your solution come from?

On edit
After some snooping, I found that solutions of this problem exist on various sites that can be made available to me for a modest monthly subscription fee. I was able to view the problem description (free of charge) in two of the sites and both had the same number for a₀ but they also included a top speed of 60 m/s which is not included in your pdf.

The answer to the title of this thread is "No because you need to know the top speed."

P.S. To see if a problem is posted on the web, enter a characteristic excerpt between quotation marks. I entered "A good model for the acceleration of a car" and the sites popped up. Isn't AI lovely?

 

[Herman Trivilino]

I think your solution to (b) will confuse your students to the point where they will give up. Don't be surprised if they ask "Will this be on the test?"

And if you say yes they will simply try to memorize their way through it with no understanding of the physics.

If you do indeed decide to go forward with assigning this problem, the way I recommend you present the solution is by starting with a graph like the one you drew. Then for $a_x$ write $\frac{dv_x}{dt} = a_o - kv_x$. Tell them you have to guess that the solution is $v_x=v_{max}(1-e^{-kt})$, and use the graph to explain why that solution seems reasonable. (Note that in your solution you are anyway guessing!) Then go on to demonstrate, by taking the time derivative and making the appropriate algebraic substitutions, that it is indeed the solution to the original differential equation.

As an instructor you can contact the publisher's rep and get a copy of the instructor's solutions manual.

Why are you using a college-level textbook in a high school class?

The solution involves an understanding of differential equations. These students are only beginning to grasp introductory calculus.

I don't think this is an appropriate problem to assign to high school physics students.

Edit: Concerning your solution to (c) I would not complicate things by introducing $\tau$. I would simply write $\dfrac{v_x}{v_{max}}=1-e^{-kt}$, substitute $0.95$ for $\dfrac{v_x}{v_{max}}$ and solve for $t$. Of course, you will not get a numerical answer. Perhaps the answer that will make the most sense to the students is $t=k \ \ln 20$.

 

[kuruman]

Don't be surprised if they ask "Will this be on the test?"

Or more likely, "Do we have to know that?"

I don't think this is an appropriate problem to assign to high school physics students.

I tend to agree. The idea that the rate of change of the velocity is proportional to the velocity is not the vehicle to teach high-school students the decaying exponential. It is just a convenient and easy-to-handle model.

I think that a natural place to teach the decaying exponential in high-school is when doing radioactivity. The idea that the decay rate is proportional to the number of undecayed nuclei is kind of obvious. If not, it can be explained as the reverse of population increase wherein the rate of change of the number of people is proportional to the number of people who give birth. Auxiliary plots with number of half-lives on the abscissa and with $~e^{-n}~$ and $~1-e^{-n}~$ on the ordinate would be useful for driving the point home.

 

[hutchphd]

I think that a natural place to teach the decaying exponential in high-school is when doing radioactivity.

I disagree. A much easier, cheaper, and more accessable model is the discharge of a capacitor. Volts is easier to measure than REM, and the invocation of something as exotic as radioactivity to this bedrock notion is counterproductive. Other examples abound.

 

[Herman Trivilino]

The answer to the title of this thread is "No because you need to know the top speed."

Certainly you cannot find a numerical value, but we do know that $v_{max}=\dfrac{a_o}{k}$ where $a_o$ and $k$ are given in the problem statement: $a_x=a_o-kv_x$.

 

[Herman Trivilino]

I think that a natural place to teach the decaying exponential in high-school is when doing radioactivity.

A much easier, cheaper, and more accessable model is the discharge of a capacitor.

Either of these are good choices, but I would start with something simpler. Paul G. Hewitt's approach is my favorite. He states that at 1:00 pm a single bacterium is placed in a jar. The number of bacteria doubles every minute, and at 2:00 pm the jar is full. At what time is the jar half full? Students will offer different answers but it will be a great surprise to most that the answer is 1:59 pm. Note that at 1:58 pm the jar is one-fourth full. Display drawings of the jar at each of these times. Then suppose that some forward-thinking bacteria realize the crisis at hand and procure some extra jars for the expansion of the colony. One extra jar will be full at 2:01 pm. Add two more jars and they will be full at 2:02 pm. (The late Al Bartlett famously said the greatest shortcoming of the human race may be its failure to understand the exponential function.) This is because the above is an example of steady growth!

Students can then be led to the creation of the equation $N=2^t$, where $N$ is the number of full jars, and $t$ is the time in minutes after 2:00 pm. From there you can make this equation more general: $N=N_o\ e^{kt}$. Of course this is an example of exponential growth, not decay, so some more instruction is needed to get to examples that involve $N=N_o\ e^{-kt}$ or $N=N_o\ (1- e^{-kt})$.

As to the question of whether capacitor charge decay or radioactive decay is a better example in the physics curriculum, I would say it depends on whether you happen to be studying capacitors or radioactivity at the time. Because after all learning physics is a study of the phenomena, but instead what most students seem to be experiencing when they take a physics class is applications of mathematics. This is a critical impedance mismatch between teacher and student, and is responsible for a lot, if not most, of the difficulties instructors report having with their students' learning.

It is up to the instructor to emphasize, not just through words, but with homework and test questions, that the class is a study of phenomena.