# Newtonian relativity puzzle **URL:** \ **Category:** Miscellaneous and Personal Stuff I Must Share\ **Created:** [November 20, 2024, 2:50pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508 "2024-11-20T14:50:24Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [November 20, 2024, 2:50pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/1 "2024-11-20T14:50:25Z")
Here’s a little physics puzzle to while away some time. A car takes a trip along a straight road. For the first hour, it travels at 60 mph. Without turning, it travels at 30 mph for the next hour. 1. What is the average speed of the car? 2. For a cop standing at an unknown spot on the road, what is the expected value of the car’s speed? There’s no tricks here, it’s just simple math. Hint: the answers to the two questions is different. ---
**Author:** ![pjd](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pjd/32/3009_2.png) [@pjd](https://boards.straightdope.com/u/pjd)\ **Post date:** [November 20, 2024, 3:43pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/2 "2024-11-20T15:43:46Z")
The way i see it :- > **1.** > > car does 90 miles in 2 hours. Therefore average speed is 45mph > **2.** > > Don’t understand the question ! > I guess the cop would expect the car to be driving at or within the posted speed limit. ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [November 20, 2024, 4:10pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/3 "2024-11-20T16:10:45Z")
> [@pjd](#): > > 2. Don’t understand the question ! What speed can a cop expect to see the car passing them by at? ---
**Author:** ![colinfred](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/colinfred/32/21338_2.png) [@colinfred](https://boards.straightdope.com/u/colinfred)\ **Post date:** [November 20, 2024, 4:14pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/4 "2024-11-20T16:14:39Z")
never mind ---
**Author:** ![Gyrate](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/gyrate/32/3588_2.png) [@Gyrate](https://boards.straightdope.com/u/Gyrate)\ **Post date:** [November 20, 2024, 4:15pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/5 "2024-11-20T16:15:16Z")
The car will have covered twice as much distance in the first hour as it did in the second. So for a cop at a random spot there’s a 2/3 chance the car will be going 60. Expected value - I’m winging the calculation here, but is it (60 x 2) + (30 x 1) = 150, and 150/3 = 50mph? Or is this nonsense? ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [November 20, 2024, 4:20pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/6 "2024-11-20T16:20:50Z")
> [@Gyrate](#): > > Expected value - I’m winging the calculation here, but is it (60 x 2) + (30 x 1) = 150, and 150/3 = 50mph? Or is this nonsense? That is exactly right. ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [November 20, 2024, 4:20pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/7 "2024-11-20T16:20:52Z")
I concur with @pjd for the first question, and @gyrate for the second, assuming that “expected speed” is referring to a mean, not a mode or median. ---
**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\ **Post date:** [November 20, 2024, 4:25pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/8 "2024-11-20T16:25:25Z")
I would think the median and mode would also be the same, given a reasonable reading of the problem, which I take to be “for a cop likely to be standing at any point along the route with equal probability”. Though, sure, I would say it is a mild weakness in the problem statement that it is necessary for the reader to make such an assumption rather than have it explicitly stated. ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [November 20, 2024, 4:30pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/9 "2024-11-20T16:30:34Z")
I don’t think that’s right. There is an arbitrary number of spaces that the cop could be standing on the road. On 2/3 of them, the driver is going at 60 mph, and at 1/3, the driver is going at 30 mph. If you list each of those spaces as a separate instance, the most common speed value, the mode, is 60 mph. If you list them in order from lowest to highest, the middle value, the median, is also 60 mph. ---
**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\ **Post date:** [November 20, 2024, 4:32pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/10 "2024-11-20T16:32:13Z")
Sorry, that was unclear. I meant I expect the median and mode to be the same. You’re right the mean is different. ---
**Author:** ![Left\_Hand\_of\_Dorkness](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/left_hand_of_dorkness/32/7156_2.png) [@Left\_Hand\_of\_Dorkness](https://boards.straightdope.com/u/Left_Hand_of_Dorkness)\ **Post date:** [November 20, 2024, 4:32pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/11 "2024-11-20T16:32:39Z")
Oh, gotcha–I see! Yeah, we agree. ---
**Author:** ![Gyrate](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/gyrate/32/3588_2.png) [@Gyrate](https://boards.straightdope.com/u/Gyrate)\ **Post date:** [November 20, 2024, 4:46pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/12 "2024-11-20T16:46:54Z")
> [@Pleonast](#): > > That is exactly right. Yay! I mathed right! ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [November 20, 2024, 5:10pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/13 "2024-11-20T17:10:08Z")
> [@Pleonast](#): > > 1. For a cop standing at an unknown spot on the road, what is the expected value of the car’s speed? What does ‘expected value’ mean here? How is he measuring the speed? ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [November 20, 2024, 5:19pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/14 "2024-11-20T17:19:46Z")
> [@TriPolar](#): > > > [@Pleonast](#): > > > > 1. For a cop standing at an unknown spot on the road, what is the expected value of the car’s speed? > > What does ‘expected value’ mean here? How is he measuring the speed?   Expected value is another way to say the average value. Doesn’t matter how the speed is measured, assume perfect enough. ---
**Author:** ![TriPolar](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/tripolar/32/3008_2.png) [@TriPolar](https://boards.straightdope.com/u/TriPolar)\ **Post date:** [November 20, 2024, 5:27pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/15 "2024-11-20T17:27:17Z")
If the cop times the value from the start to the end then 45MPH is the expected value. Otherwise it’s anywhere between the maximum reverse speed and the maximum forward speed the car is capable of travelling. ETA: You did specify the operating speed of the car for each mile so unless the cop measures a time span that crosses the midpoint then it’s either 30MPH or 60MPH. If the time span measured crosses the midpoint it’s somewhere between 30MPH and 60MPH. The question also assumes the car can immediately deaccelerate from 60MPH to 30MPH at the midpoint. ---
**Author:** ![pjd](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pjd/32/3009_2.png) [@pjd](https://boards.straightdope.com/u/pjd)\ **Post date:** [November 20, 2024, 5:30pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/16 "2024-11-20T17:30:10Z")
> [@Pleonast](#): > > Expected value is another way to say the average value. Well, the average speed was asked for in part 1 and it’s 45 mph. ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [November 20, 2024, 5:40pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/17 "2024-11-20T17:40:04Z")
> [@pjd](#): > > > [@Pleonast](#): > > > > Expected value is another way to say the average value. > > Well, the average speed was asked for in part 1 and it’s 45 mph. That’s the average speed of the car over the entire trip. The average speed of the car as it passes the cop is different. ---
**Author:** ![Dr.Winston\_OBoogie](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.winston_oboogie/32/2945_2.png) [@Dr.Winston\_OBoogie](https://boards.straightdope.com/u/Dr.Winston_OBoogie)\ **Post date:** [November 20, 2024, 5:50pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/18 "2024-11-20T17:50:37Z")
> [@Pleonast](#): > > That’s the average speed of the car over the entire trip. The average speed of the car as it passes the cop is different. [pedant] No, the _ **expected value** _ of the speed of the car as it passes the cop is different.[/pedant] ---
**Author:** ![Pleonast](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pleonast/32/1183_2.png) [@Pleonast](https://boards.straightdope.com/u/Pleonast)\ **Post date:** [November 20, 2024, 5:51pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/19 "2024-11-20T17:51:35Z")
> [@TriPolar](#): > > You did specify the operating speed of the car for each mile so unless the cop measures a time span that crosses the midpoint then it’s either 30MPH or 60MPH. If the time span measured crosses the midpoint it’s somewhere between 30MPH and 60MPH. The question also assumes the car can immediately deaccelerate from 60MPH to 30MPH at the midpoint. Yep, there’s lots of quibbles that can be made. Feel free to do the math for whatever assumptions you’d like to use. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [November 20, 2024, 5:52pm UTC](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508/20 "2024-11-20T17:52:41Z")
> [@Pleonast](#): > > Expected value is another way to say the average value. Nope. _Weighted_ average. That’s why it is not equal to the mean. E(x) = Σ[xi X p(xi)] Also, it is an error to say **the** average. There are many averages arithmetic mean geometric mean quadratic mean Heronean mean median midrange etc. I’ll also point out for those discussing the mode - mode is not an average. Not sure if people were using it in that way. [Next page](https://boards.straightdope.com/t/newtonian-relativity-puzzle/1010508.md?page=2)