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Dmitri Krioukov (2012), The proof of innocence, arXiv:1204.0162

Below are some of the best students answers to the summary questions. This link takes you to a PDF document of the answers (with the maths and non-maths answers all combined together). The answers to the non-maths path are also presented below.

SUMMARY QUESTIONS

Explain why an object moving at a constant speed does not necessarily appear that way to an observer as it moves towards, past and then away from a stationary observer.

An observer’s perception of an object’s speed depends not only on the actual velocity but the angle subtended by the object’s path. Things closer to an observer appear larger, so despite the fact that the object travels equal distances in a given time interval (due to its constant speed), this distance travelled appears larger when the object is closer to the observer. As a result, when it is closer to the observer, the object appears to ‘sweep out’ a larger angle in a given time. This means that the angular velocity of the object - from the observer’s reference point - also increases as the object nears the observer. Angular velocity = angle/time.

Yen Li

When an observer measures the speed of an object passing by, they will not be standing directly facing that object usually so angular rather than linear velocity is measured. In the author’s case, the police officer is standing at an angle to the car, changing their whole perspective on the event. For the speed to remain constant, the angular speed and the radius must be inversely proportional to one another demonstrated by the equation v=ωr. This means that as the vehicle is moving towards the observer, the radius will be decreasing which results in an increase in angular speed so the car appears to be moving faster. When the car has just past the observer, the radius will be at its lowest so the visual speed is maximized. Finally, as the vehicle moves away, the radius will be increasing meaning that the car appears to be decreasing in speed even though the vehicle has been moving at a constant speed the whole time.

Katie

In a similar way to my notes for section II. Constant Linear Speed, send in your notes on how you arrived at the solution for section III. Constant Linear Deceleration And Acceleration. You can submit this as a photo of your hand-written notes if that’s easier – however suits you best.

Kitty gives a detailed and well-explained derivation of the formula that relates the distance to the acceleration.

Kelsey provides a very neat and complete solution to the entire derivation (I've added one note on the second page to point out that we don't need two integrations).

As budding physicists, do you think that this paper presents an adequate defence for the author’s position that he did actually stop at the stop sign?

While mathematically valid, the paper not only relies on a few assumptions but what the author has proved is not actually that he stopped at the stop sign but that the observer could not have known whether he stopped or not- this gives him the benefit of the doubt but does not prove that he truly stopped, for which the reader only has the author’s own testimony.

Angela

Some significant assumptions are made by the author, including the fact that C₂ travels at a constant velocity and does not stop, despite also passing the stop sign. (However, this is potentially due to C₂ being on a separate road parallel to that of C₁, or travelling in the opposite direction). Additionally, as addressed in the paper, the average acceleration of a Toyota Yaris is 1.77 ms^-2 (100 kmh^-1 in 15.7 s) whereas the model uses an acceleration of 10 ms^-2. A deceleration of this magnitude would perhaps be possible, particularly due to the author’s purported sneeze, but the Yaris would certainly struggle to accelerate that quickly after coming to rest. Additionally, if the acceleration were not this high (as is likely), the graph of the angular velocity would have its two maxima further apart and outside the t_p window; this would have made it clearer that the defendant had been slowing down and thus the policeman would have been unlikely to issue a fine in the first place.

Yen Li

The author states that a “Yaris accelerates to 100 km/h in 15.7 s”, but claims that an acceleration of 10m/s² in the early stages may be justified. I have also seen that a Toyota Yaris moves from 50mph to 70mph in around 8s. Do you think it’s possible that an acceleration of 10m/s² is achievable for a Toyota Yaris, given the manufacturer’s data?

Ambre presents a very interesting answer in his solutions - available to download here. He uses a model based on the engine delivering a constant power and finds that the Yaris can achieve an acceleration of 10m/s² but only for a very short period of time.

This week we also gave a second article on atomic clocks, Jared has come up with SKIM-READ QUESTIONS (with answers) and SUMMARY QUESTIONS (with answers) that can be seen here.

To help this week, we have a series of tools that might be useful:

• We’ve made this applet to illustrate dynamically the two situations that the author describes. You can alter the acceleration of the car and the distance away the observer is to see (from the arrows) how the angular speed changes.
• You might want to use an online function plotter to try and plot some functions for yourself. Do email if you need any help.
• You also might want to check any integration and differentiation makes sense using an online tool.

With all of the tools above, don’t rely on them to do the work for you. You should be able to attempt all of this plotting and calculus with pen and paper, but you should use them to check yourself.

 

1. INTRODUCTION

 

Why does a stationary observer not measure the speed of an object?

Because of your perspective. Things further away look smaller and take up a smaller amount of your field of view. As they move, they don’t traverse much of your field of view to begin with so seem to travel slowly. When an object is moving directly past you, it traverses a lot of your field of view in a short time so seems to be moving faster. Consider it in terms of how much you need to turn your neck as an object moves past you left to right. Your neck has to turn most quickly when the object passes your position.

What scenarios is the author going to consider?

A car travelling at a constant speed (and hence driving through the stop sign without stopping). A car decelerating rapidly to a halt at the stop sign and accelerating away from the stop sign again rapidly (in the same direction as they were previously travelling).

 In the next section, the author will choose to use the point when the car is at the stop sign (when C is at S) as the origin of time – the point when t=0s. Dealing with negative time is always a little confusing, but importantly for the argument running through this paper, all of the scenarios will be in some sense symmetrical about t=0s. You can understand all of the maths by simply considering positive values for time (and this is sometimes beneficial).

I. CONSTANT LINEAR SPEED

This link gives an example of how I would take notes and convince myself of all the steps in the working out through a paper. I’m annotating all of the steps, clarifying why things are the way they are and filling in the intermediate steps to check the maths.

If you want to see proof of the differentiation of arctan(t) this link shows how you can derive it.

Why is equation (2) not technically correct?	Because x can’t be negative – it’s a length. If t<0 then equation (2) would give negative values for x. Technically, for this situation we should probably write x(t)=v0|t|
What are the benefits of the author choosing the time t=0s to be the point when the car meets the stop sign?	It makes it easier to write symmetric functions around the origin. It’s also the only real fixed point in time in the entire scenario we’re looking at. All of the other ‘moments’ depend upon how fast the car is going – it doesn’t start or end in a definite place. But the stop sign is a definite location and the observer’s view of it gives a definite time point.
What rule of differentiation does the author use in Equation (6)?	Chain rule.
Describe the graph shown in Figure 2. Link it to what is physically happening in the example of the car moving at a constant speed.	When the car is approaching the stop sign, the angular speed is initially small as the car is far away. As the car approaches, the angular speed increases. The angular speed is maximum when the car reaches the stop sign (which is the point t=0s). As the car moves away, the angular speed decreases. When the car is far away once again, the angular speed becomes small.

 III. CONSTANT LINEAR DECELERATION AND ACCELERATION

 Remember, we have made this applet to help you visualise the difference in motion.

Explain the steps the author takes between equation (9) and (12) to prove the relationship given in equation (8)	Equation (9) gives the velocity in a situation of constant acceleration in a situation where v=0 when t=0. Equation (10) defines velocity as the time differential of position. This can be rearranged, in equation (11), as a derivative can be understood loosely as a fraction of dx/dt=little bit of distance / little bit of time and so dx and dt can be manipulated as in normal algebra. Equation (12) starts by stating that position is the integral of lots of small bits of position up to the point we care about. You’ll notice the limits of the integration here are between 0 and t, allowing us to not have to think about negative time. By replacing dx with equation (11) and changing the limits to reflect that we’re now looking at the time equivalents (t=0 when x=0 and by definition t=t when x=x). As we’re working with constant acceleration, we can replace v with that in equation (9). Performing the integration gives us the last step.
Using the explanation given after equation (12), can you derive the relationship given in equation (13)?	This is best seen in the PDF version of the answers - link.
Describe the graph shown in Figure 3. Link it to what is physically happening in the example of decelerating/accelerating car.	When the car is far away and approaching, the angular speed is small. As the car approaches, the angular speed increases. As the car approaches, though, it is also decelerating – this means that the maximum angular speed is reached before the car passes by. Once the maximum angular speed is reached, the angular speed decreases to zero. At the point when the angular speed is zero, the car has stopped at the stop sign.   The car then accelerates rapidly, and the angular speed increases rapidly. This is balanced, though, by the fact that the car is also moving further away, which has a diminishing effect on the angular speed. These effects combine so that the angular speed reaches a maximum before decreasing. When the car is far away (and still accelerating), the angular speed is small again.

IV. BRIEF OBSTRUCTION OF VIEW AROUND

 Remember, we have made this applet to help you visualise the difference in motion.

What are xf and xp?	xf is the distance over which the author’s car is fully obscured by the longer car. xp is the distance over which the larger car at least partially obscures part of the author’s car.
What is the biggest assumption made in this section?	That the car can achieve an acceleration of 10m/s2.
What is the time t' ?	The time taken for the angular speed to reach its maximum in the situation when the car has stopped at the stop sign and then accelerated quickly.
What conclusion does the author draw from the fact that tp>t' ?	That his car could have been partially obstructed by another for longer than it took for him to accelerate to a

V. CONCLUSION

Describe how the three bullet points in the conclusion link to Figure 5.

By measuring angular speed, your observation is strongly focussed around the moment when an object passes you to understand how it is moving – if an obstruction happens in this moment, your brain is forced to extrapolate. In both scenarios shown in figure 5, the curves are changing most rapidly around t=0s and are very similar at t>2s and t<-2s.   By decelerating and accelerating quickly, the angular speed over time is significantly altered, forming a peak either side of t=0s – shown by the blue line in Fig. 5. But, given the abrupt nature of the acceleration and deceleration, these peaks are fairly close to one another, so that it does bare some resemblance to a constant velocity scenario – shown by the red line.   By estimating the window over which the view of the observer was at least partially obstructed – shown by the black dotted lines - the author has been able to show that such an obstruction, combined with such a sharp deceleration/acceleration could be misinterpreted easily. This observer misinterpretation is shown by the dotted red line.

 FURTHER READING

This week, we don’t have further reading tied to the topic, just a completely separate (and short) article from Physics Today entitled Transportable clocks achieve atomic precision about using portable atomic clocks to measure gravitational redshift. This is an excellent opportunity to practice writing your own skim-read and summary questions. Feel free to send in your work!

GENERAL INFORMATION

Remember, reading a paper isn't like reading a piece of fiction or a newspaper article. Don't get frustrated if it doesn't immediately make sense - you might need to do a little research of your own to understand some of the ideas. This article gives you an idea of how scientists read differently.

Each question refers to a specific part of the paper e.g. Page 2, Column 3 is written as (P2, C3).

Next week, we'll publish solutions to the questions and the best submitted summaries from students across the country.

NEXT WEEK

We're going to explore how physics interacts with other fields. Specifically, we're going to look at the physics of living systems. Within this link, look through the editorial and the first comment - Biophysics across time and space. This link gives an overview of the different aspects of biophysics that are currently studied.