# 1000m race

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Homework Statement:
In 1 km races, runner 1 on track 1 (with time 2 min, 27.95 s) appears to be faster than runner 2 on track 2 (2 min, 28.15 s). However, length L2 of track 2 might be slightly greater than length
L1 of track 1. How large can L2 - L1 be for us still to conclude that runner 1 is faster?
Relevant Equations:
Vavg = total displacement/ total time
First I calculated the avg velocity with 1000m for both runners. It came 6.76m/s and 6.75m/s. It suggests that the velocities are same. This means yes that the L2 track is slightly longer than L1.
Then why is it asking this question (…that runner 1 is faster)? Both are at equal speeds.
I don’t get the question really.
If we really want the runner1 to be faster we need to shorten his track.

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I don’t get the question really.
Suppose L2 is 1.1 km. who would be faster ?

##\ ##

Steve4Physics
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Suppose L2 is 1.1 km. who would be faster ?

##\ ##
Runner 2 with 7.4 m/s vs 6.8 m/s

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Both are at equal speeds.
Really? Does 147.95s = 148.15s? The times have 5 significant digits so you cannot assume that 6.76m/s=6.75m/s

Let L2-L1=x; x≥0
For what values of x is the first runner faster?

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Really? Does 147.95s = 148.15s? The times have 5 significant digits so you cannot assume that 6.76m/s=6.75m/s

Let L2-L1=x; x≥0
For what values of x is the first runner faster?
Let me elaborate the question. Runner 1 appears to be faster. But both are running at same speed. The track 2 is longer or might be longer(idk) but it should be longer if they have to run at same speed. I guess that’s how they are made. More curvature and shorter track vs less curvature and longer track. Now how large the L2 be than L1 that we can still conclude that runner 1 is faster. Is it actually faster or appear to be faster?

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O

Ok! But how will we solve this?
Often, I find the hardest part of the problem is picking a good set of variables so that one can write down some good equations that are easy to solve.

You pick up some tricks over the years. One trick is that when you see an inequality ("less than", "greater than", "at least", "at most", etc), you want to look for the boundary. The boundary is an equality. You can write down an equation for it. Equations are good. You can solve them.

So when the problem talks about one runner's velocity being greater than the others, you know almost immediately that you are going to be writing down an equation that is basically:$$v_1=v_2$$So take out a piece of paper and at the top of it write down that equation.

Now, ##v_1## is ##\frac{L_1}{t_1}## and ##v_2## is ##\frac{L_2}{t_2}##. Update your equation accordingly.

The problem hints that you can take ##L_1## to be fixed at 1 km exactly and vary ##L_2## so that the equality is satisfied. That leaves you with one equation and one unknown.

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BvU, neilparker62, jim mcnamara and 1 other person
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So... are we to guess which track (if any) is actually 1km long ?

jbriggs444 and Lnewqban
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Often, I find the hardest part of the problem is picking a good set of variables so that one can write down some good equations that are easy to solve.
Let me elaborate the question. Runner 1 appears to be faster. But both are running at same speed. The track 2 is longer or might be longer(idk) but it should be longer if they have to run at same speed. I guess that’s how they are made. More curvature and shorter track vs less curvature and longer track. Now how large the L2 be than L1 that we can still conclude that runner 1 is faster. Is it actually faster or appear to be faster?
It’s confusing!

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The problem seems poorly stated.
They mix times and distances that should be both 1000 meters, but suddenly are not.
Who has measured those times and between what two marks that contain 1000 meters in eack track?

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It should be easy to see that shortening track L1 by a little will have approximately the same effect on the margin of victory as lengthening track L2 by the same amount. The speeds of the two runners are very nearly equal, so the faster one will not be extending his lead by very much over the last little distance.

In the interest of having a definite problem to solve though, fix ##L_1## at 1 km and let ##L_2## vary, state this assumption and move on. If, after solving the problem, you want to go back, fix ##L_2## at 1 km, and solve for ##L_1##, please feel free to do so and see whether the result for ##L_2-L_1## is approximately the same.

Steve4Physics and hmmm27
Staff Emeritus
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First I calculated the avg velocity with 1000m for both runners. It came 6.76m/s and 6.75m/s. It suggests that the velocities are same. This means yes that the L2 track is slightly longer than L1.
I don't get your logic here. You're saying if you assume L1=L2, the speeds are the same; therefore, L2>L1 which seems to suggest you're saying the speeds aren't the same. Which is it?

Gold Member
Homework Statement:: In 1 km races, runner 1 on track 1 (with time 2 min, 27.95 s) appears to be faster than runner 2 on track 2 (2 min, 28.15 s). However, length L2 of track 2 might be slightly greater than length
L1 of track 1. How large can L2 - L1 be for us still to conclude that runner 1 is faster?
Relevant Equations:: Vavg = total displacement/ total time

First I calculated the avg velocity with 1000m for both runners. It came 6.76m/s and 6.75m/s. It suggests that the velocities are same. This means yes that the L2 track is slightly longer than L1.
Then why is it asking this question (…that runner 1 is faster)? Both are at equal speeds.
I don’t get the question really.
The problem doesn't say that runner 1 is faster than runner 2.
It says that runner 1 won the race with a better time.
You are to find out if he is actually faster in this race, by challenging if it was a actual fair race if the track lengths are not equal.

You can make 3 cases,
1. Both track length are equal. Then clearly runner 1 is faster.
2.Track length 1 is longer. Then clearly runner 1 is faster,
3 Track length 2 is longer, Then we are not sure who is fastest in this race and will have to do some math to find out if a possible error in track length could be a reason for the times.

assume that far each and every race before this particular one, both runners ( their clones ) clocked in at time T, in an exact tie and both win gold.
But for this race runner 2 clocks in behind runner 1.
Thus, all other things being equal, runner 2 must have had to run a longer distance if he is just as fast as runner 1.

So either track length 1 is shorter than 1000m, and track 2 is exactly 1000 m. ie L2>L1
Or track length 1 is exactly 1000m, and track 2 is longer than 1000m. ie L2 > L1
Or, neither is 1000 m but differ in length, with L2>L1.

You can go through the possibilities of how much L2>L1since you know the clock times.

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But both are running at same speed.
You do not know that. Your calculation to that effect in post #1 was flawed in two ways:
1. You assumed they both ran 1km exactly, even though you are told that may not be true.
2. You did not keep enough precision for the small difference in their times to result in a difference in speed.

However, the question asks you to find the length difference that would completely account for the time difference, i.e. on the assumption that they did run at the same speed.

Gold Member
You are to find out if he is actually faster in this race, by challenging if it was a actual fair race if the track lengths are not equal.
So we are given that the length of tracks are not equal and we are to find out how much can L2 be greater/vary than L1 still to conclude that runner 1 is faster. If it’s greater than this value then runner1 is not faster and the race is not fair.
@jbriggs444 I have done the solution. And it turns out L1-L2 < 1.35 m.

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So we are given that the length of tracks are not equal and we are to find out how much can L2 be greater/vary than L1 still to conclude that runner 1 is faster. If it’s greater than this value then runner1 is not faster and the race is not fair.
@jbriggs444 I have done the solution. And it turns out L1-L2 < 1.35 m.
( Actually any race with not equal track lengths is unfair )

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So we are given that the length of tracks are not equal and we are to find out how much can L2 be greater/vary than L1 still to conclude that runner 1 is faster. If it’s greater than this value then runner1 is not faster and the race is not fair.
@jbriggs444 I have done the solution. And it turns out L1-L2 < 1.35 m.
- find Runner-1's speed (assuming L1 = 1000m);
- find the extra time Runner-2 runs;
- assuming the runners have equal speeds, ask yourself how much extra distance Runner-2 covers in the extra time.

256bits
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- find Runner-1's speed (assuming L1 = 1000m);
- find the extra time Runner-2 runs;
- assuming the runners have equal speeds, ask yourself how much extra distance Runner-2 covers in the extra time.
Strictly speaking, we don’t know which track is exactly 1km, if either. So all we can calculate is the ratio of lengths that would account for the ratio of times,

Lnewqban
Gold Member
( Actually any race with not equal track lengths is unfair )
The person on the inner circular track has to cover the least distance vs the person on outer tracks. So the person on the inner track sit very behind from everyone else. The person on outer track sit forward.
But we shouldn’t forget about the curvature which takes toll on runners speed. Shouldnt we must sit equally.

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256bits
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The starting point for each racer are different in circular races. The one with smallest circle starts from the very behind. The one on outer circles start from very forward. However, with more curvature they run a less distance vs with those who have less curvature.
Are those races in circle have equal length tracks?
They should have equal distances lanes if the lanes were stretched out linearly.
I don't think they take into account the fact that it is easier to run on a track more straight than curved, where the more inner guy has to have a bit more sideways step in the run.
some tracks are angled to compensate but not all.

Maybe they do, but I wouldn't count on it.
good question

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The person on the inner circular track has to cover the least distance vs the person on outer tracks. So the person on the inner track sit very behind from everyone else. The person on outer track sit forward.
But we shouldn’t forget about the curvature which takes toll on runners speed. Shouldnt we must sit equally.
We normally ignore any effect from a tighter curvature over a shorter distance. Without data to indicate how much disadvantage (or advantage) this causes, we cannot correctly handicap a race to account for such curvature by adding or subtracting distance.

What we do know for certain is that starting all the runners even is the wrong thing to do. That gives the inner track a significant advantage.

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We normally ignore any effect from a tighter curvature over a shorter distance. Without data to indicate how much disadvantage (or advantage) this causes, we cannot correctly handicap a race to account for such curvature by adding or subtracting distance.

What we do know for certain is that starting all the runners even is the wrong thing to do. That gives the inner track a significant advantage.
You know if bicycle races take into account the added potential energy of the rider in the top lane.

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Strictly speaking, we don’t know which track is exactly 1km, if either. So all we can calculate is the ratio of lengths that would account for the ratio of times,
Agreed. But I was replying to @rudransh verma's Post #14. @rudransh had applied @jbriggs444's pragmatic approach (from Post #10):
In the interest of having a definite problem to solve though, fix ##L_1## at 1 km and let ##L_2## vary, state this assumption and move on.

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We normally ignore any effect from a tighter curvature over a shorter distance. Without data to indicate how much disadvantage (or advantage) this causes, we cannot correctly handicap a race to account for such curvature by adding or subtracting distance.
So when we are in doubt whether we should include or not to make it a just and fair race we just ignore it. This is the way science goes?

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We normally ignore any effect from a tighter curvature over a shorter distance.
Who are "we" here?
I thought race organisers went to some lengths, so to speak, to ensure the distances were the same for all lanes.

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Who are "we" here?
I thought race organisers went to some lengths, so to speak, to ensure the distances were the same for all lanes.
Right. Distances are the same. Presumably as measured with a tape along the inner lane line. The tighter radius of curvature in the inner lanes is ignored.

You don't get a lowered distance to compensate for increased dizziness, for instance.

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So when we are in doubt whether we should include or not to make it a just and fair race we just ignore it. This is the way science goes?
If you can measure its effects then you can include it. If you cannot measure the effects then it will be difficult to include in your model.

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As stated, the exercise not only does not specify a track with curved lanes, but doesn't even specify if it's the same track.

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=
As stated, the exercise not only does not specify a track with curved lanes, but doesn't even specify if it's the same track.of
In fact the original question does refer to "track 1" and "track 2" - so we can assume they are different tracks. But, more worryingly, we are not told if the two tracks are in the same inertial frame of reference.

For simplicity, but retaining some degree of generality, I suggest assuming that:
- track 1 is in frame 1 and track 2 is in frame 2;
- the proper length of track 1 is 1000m;
- the tracks are straight and co-linear (along some line, L);
- the relative velocity of frame-2, as measured by a stationary observer in frame-1, is ##\vec Ξ##;
- the two given time-measurements, ##L_1## and ##L_2## are all measured by a 3rd observer who is moving along L but is stationary relative to the centres of mass of the 2 runners (for convenience, we may assume the runners have equal rest masses).

This should then render the problem more manageable.

hmmm27
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[...] we may assume the runners have equal rest masses).
You may be over-thinking this.

Steve4Physics
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In fact the original question does refer to "track 1" and "track 2" - so we can assume they are different tracks. But, more worryingly, we are not told if the two tracks are in the same inertial frame of reference.
The problems in resnik are written in a very concise manner. So sometimes questions which are tricky confuses me and it takes quite an effort to understand the situation (what is given and what eqns can be made). It should be elaborate.
But there is no need for inertial frames and all that since it’s just a question from chapter 2 motion in 1D.

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You may be over-thinking this.
Isn’t this what physics is about😨?
We are overthinker and that’s a hurdle in solving simple problems.

Steve4Physics
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Isn’t this what physics is about😨?
We are overthinker and that’s a hurdle in solving simple problems.
Just to clarify something (if clarification is needed)…

My post #28 (about the possibility that the tracks are in different inertial frames of reference) was supposed to be a joke! (Well, I thought it was funny.)

It was intended to be a humorous illustration of what happens when overthinking goes too far. But maybe it failed!