Was using distraction tactics to keep my brain away from thinming about the pain on a recent attempt at a PB on my 40km ride home the other day.

Made up a question to keep me occupied.

I couldn’t work it out but maybe you can!

A cyclist enters a bike race. It is 30km long. The first 10km is up a steep hill. He averaged 10km/h. He (or she) crests the hill and hits terminal velocity immediately and completes the race with his (or her) strava starting that the average speed for the ride was 30km/h.

How fast would he (or she) have to be going in the last 20km yo get that average and how long did the whole race take?

Over to you!

blooooooody fast

He or she would have had to teleport from the crest of the hill to the finish.

Infinite. Edit: or of course teleport as stated above.

he has spent an hour doing just 10k. He only has an hour to do 30k.

Yep.

Instantaneous teleportation is the only answer.

10 km/hr for the first 10 K = 1 hour

30km/hr for full 30k = 1 hour

Time left to complete last 20k = 0

Well to give the cyclist some chance of pulling it off, if you assumed the 10 km/h was their average rounded to the nearest whole number; they could just about manage it if they did over approximately 428 km/h (I think??).

But surely he can still average 30kph over 30km even if he was longer than an hour.

For example if you travelled for 60km in 2 hours your average is 30kph. If you travel 15km in 30mins your average is 30kph

So if he did the last 20km st, say 100kph what would be the overall average??

He would have done 30k in 72 minutes. 25kph

Unless he is in England. Where we use imperial measurement.

What speed for the last 20km??

10k in 1hr,

Total distance 30km

Average speed 30km/h

Assuming this isn’t completely made up and you’re cross about someone else’s Strava time on your segment…

Its a trick question

The average speed is based on distance covered not time taken. (i.e. its an average of speeds v1:vn/n, not the overall speed d/t)

So 20km are covered at 40km/h

Total distance covered 30km @ 30km/h average

30×30 = 10×10 + 20×?

?=(900-100)/20 =40

What speed for the last 20km??

You said 100kph. So 20k takes 12 minutes.

Well that beats my commute, I was trying to remember all the lyrics to “A land down under” but kept getting stuck at “vegemite sandwich”

The average speed is based on distance covered not time taken.

It’s based on both surely ?

The point is, if you have an average speed of 30km/h over a ride, the ride cannot have taken more than 1hr if the total distance is 30km.

You have to have taken exactly 1hr to complete the 30km to have an average speed of 30km/hr on a 30km ride.

So teleport is the only option 👍

Well that beats my commute, I was trying to remember all the lyrics to “A land down under” but kept getting stuck at “vegemite sandwich”

I spent my last one singing Baby Shark to myself. I think hypothesized that Tom Boonen doesn’t likely doesn’t do this when he’s on his bike. Then I went back to Baby Shark anyway.

Was this on a conveyor belt?

So  you are saying that if I ride 60km I cannot feasibly achieve an average of 30kph??

EDIT: Scrub that! Looked at the OP and saw 40km ride not 30km.

The ride is 30km though!

 So  you are saying that if I ride 60km I cannot feasibly achieve an average of 30kph??

No. not saying that at all.

But if your ride is 30km. And you spend an hour doing the first 10k. Then an average speed of 30km/hr is not possible.

Similarly. With your 60km ride, if you spend 2 hours doing the first 20km, then an able rage speed of 30km/hr is not possible.

It’s based on both surely ?

The point is, if you have an average speed of 30km/h over a ride, the ride cannot have taken more than 1hr if the total distance is 30km.

You have to have taken exactly 1hr to complete the 30km to have an average speed of 30km/hr on a 30km ride.

So teleport is the only option 👍

I’d expect it to be basic d/t so mean “actual” velocity for the ride.calling average makes it a bit circumspect though, in theory the average velocity would be an average of velocities not the overall calculated one. I’d really not expect that level of distinction though in the same way I’d expect average to be mean not mode or median unless someone made it explicit.

My maths is, I’m fairly conifdent, correct and gives a mean average speed over any 1k* of 30km/h but does require deliberately “misinterpreting” the question, i.e. assuming there IS a solution.

If usual norms apply, no it’s impossible.

So  you are saying that if I ride 60km I cannot feasibly achieve an average of 30kph??

No one said that, what people said is you can’t do 30km in a specified hour if it took you that same hour to do 10km.

It’s mathematically only possible if you calculate the average speed in an odd way.

But surely he can still average 30kph over 30km even if he was longer than an hour

Nope, if your distance is capped at 30km and it takes over an hour you have failed to average 30kph.  The only way to reach an average of 30kph is to extend the distance.

So 20km are covered at 40km/h

You saying that covering 30km in 1.5 hours is an average speed of 30kph?

So you are saying that if I ride 60km I cannot feasibly achieve an average of 30kph??

Yes you can so long as you complete the ride in exactly two hours.

This is why XC race bikes suck as trail bikes. If you fall behind on a climb, you have to be massively faster on the descent to make up the time. Therefore, a bike/rider that climbs well but descends poorly will beat a bike/rider that descends well but climbs poorly. A bike optimized for highest average speed will therefore suck on descents, so will not be much fun for most people.

I vote time travel.

As others have said, overall ride average is 30km/h, so for a 30 km ride you must ride it in exactly one hour. The problem is that you’ve already taken one hour to do the first 10km, so it’s technically impossible to achieve a faster average speed than 30km/h.

You can see that the result trends to 30km/h average as speed increases, but will never reach it unless your speed for the second section is infinite – i.e. zero duration to travel the last 20km.

average velocity would be an average of velocities

Just to be a pedant… we’re talking speed here, not velocity

You saying that covering 30km in 1.5 hours is an erage speed of 30kph?

Yes and no…

30/1.5 =/= 30

But…

If i said

10km/h per km for d and 40km/h for 2d, whats the average value of v for any given subdivision of d, the answer is 30km/h.

The average speed worked out that way is 30km/h.

The issue is you know that v=d/t, you know the rational way to calculate the average value of v is total distance / total time. On that basis its impossible to achieve 30kph. You don’t want to work out v/d because its “wrong”.

It’s deliberately manipulating the question to give an answer. But if it were apples per km you’d have no problem with working it out like like that. 10apples/km for 10km how many apples in the next 20km to achieve an average of 30 apples per km? You know v is a function of time so you’re developing from that.

Sure but the question was

A cyclist enters a bike race. It is 30km long.
The first 10km is up a steep hill. He averaged 10km/h.
the average speed for the ride was 30km/h.

How fast would he (or she) have to be going in the last 20km yo get that average and how long did the whole race take?

and your answer is?

and your answer is?

It depends how you calculate it.

Option 1) modal speed, he does 30km/h for 2hours, (or 20km if you want to use distance) the modal speed is 30km/h. Its not wrong its just you’ve got to do it in the not obvious way.

Option 2) median speed, as above.

Option 3) mean speed (because no one uses median and mode)…

At no point does the question say it takes an hour, you think it does because you think average speed is total distance over total time. Its not. Mean speed is sum of values of speed divided by number of values of speed so…

(10km/h * 10km + 40km/h * 20km)/30km = 30km/h

Equally

(10km/hr *1hr + (<1/0)km/h * 0hr) =/= 30km/h (but you can’t do this with the information in the question, you’re inferring the time)

But given the question presupposes it is possible, the anwser has to be it didn’t take an hour.

I think in likelihood the question is wrong but its not impossible to solve.

What was at the top of the hill that suddenly made him consider gender neutrality?

But given the question presupposes it <strong class=”bbcode-strong”>is possible, the anwser has to be it didn’t take an hour.

I think in likelihood the question is wrong but its not impossible to solve.

I’m going with angry tweets to the exam board here!! If your getting that deep into the question then it’s going to be a technicality answer.

I’m right. Suck it up and go home. Hopefully it’s not more than 30k away.

It depends how you calculate it.

Option 1) modal speed, he does 30km/h for 2hours, (or 20km if you want to use distance) the modal speed is 30km/h. Its not wrong its just you’ve got to do it in the not obvious way.

There’s no such thing as modal speed in something that has infinite variation.

Option 2) median speed, as above.

Median (middle value) speed is possible, but there’s no way this was the intended answer because “average” is almost always used to mean…

Option 3) mean speed (because no one uses median and mode)…

At no point does the question say it takes an hour, you think it does because you think average speed is total distance over total time. Its not. Mean speed is sum of values of speed divided by number of values of speed so…

(10km/h * 10km + 40km/h * 20km)/30km = 30km/h

Err, there’s some very dubious maths there. Allow me to present it more correctly:

10km @ 10km/h == 1 hour of travel
20km @ 40km/h == 30 minute of travel

Average speed is 30km/1.5 hours = 20km/h

Equally

(10km/hr *1hr + (<1/0)km/h * 0hr) =/= 30km/h (but you can’t do this with the information in the question, you’re inferring the time)

0km/h for 0hrs means he didn’t move, so he averaged 10km/h for the 10km he did, and didn’t ride the rest – I’ve claimed rides like this myself before now!

Can I cheerfully point you at my graph above which summarises this quite nicely? You can’t do 30km/h average (inferred “mean”) speed with the information provided, so the question is incorrectly presented.

Err, there’s some very dubious maths there. Allow me to present it more correctly:

10km @ 10km/h == 1 hour of travel
20km @ 40km/h == 30 minute of travel

Average speed is 30km/1.5 hours = 20km/h

Extremely dubious I’d say. Average speed is total distance/total time anything else would not make sense to me. [Someone may know different for close to light speeds I perhaps].

There’s no such thing as modal speed in something that has infinite variation.

Mode simply requires a series of values, so if the values taken are every 5minutes, or the speeds of 50 riders over the course or what ever doesn’t matter, if you’ve got a series of values you can find a modal one, regardless of what those values represent. But regardless it doesn’t matter, the chances of the question requiring you to populate a fictional range with assumed values are about as close to zero as the chances of pedaling 20k in 0 seconds.

because “average” is almost always used to mean…

Completely agree and I’ve said as much (but oddly the accepted meaning of “average speed” isn’t the mean)

0km/h

Isn’t the same as 1/0km/h but that’s not really at debate.

Err, there’s some very dubious maths there. Allow me to present it more correctly:

I in no way disagree, the dubious-ness isn’t in the maths though, it’s in the wording. Which I’ve openly admitted I’m deliberately stretching to get to an answer.

Speed is, without question, distance over time. You (and I for that matter) would generally refer to the average speed as the total distance over time, (d1+d2)/(t1+t2)etc.

And average of speeds would be  sum(s1:sn)/n

If you had 50 riders doing the same race. 20 taking 1hour (30kph) 10 taking 1.5(20kph), 20 taking 2hours(15kph) would you calculate the average speed of a rider as

A) 50×30km/(20*1h+10*1.5h+20*2h)= 16.7km/h

B) or as (20*30kph+10*20kph+20*15kph)/50h =22km/h.

I’d go for B (and i think I’d be in the majority there), i wouldn’t normally use that method for a single rider, not least as you tend not to have the information but the only way to get an answer that isn’t 1/0 (or 20/0 if you’d rather) is to not treat speed as a special case and rather to work out the average speed as a mean. As if, fer instance, the rider were flashed by a camera at 1/3rd, 2/3rd & end, with no given value of t or d you would simply take the mean of those three values and be done with it.

As in at least one of my replies, the question is bad but  its mathematically possible, you just have to decide exactly what you’re showing and you have to stretch to do it, by ignoring accepted norms (i.e. average speed really just being speed, which is an average).

anything else would not make sense to me. [Someone may know different for close to light speeds I perhaps].

I don’t think you’ll find many dissenters on that. It doesn’t make sense to do it any other way.

and your answer is?

It’s not possible, as explained above multiple times.

Average speed = total distance / total time.

That’s what it means. If you think it means something different, you don’t understand basic maths and physics.