Kinematics of three runners in a race

  • Thread starter bkraabel
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  • #1
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Homework Statement



Runners A, B, and C run a 100-m race, each at a constant speed.
Runner A takes first place, beating runner B by 10 m. Runner
B takes second place, beating runner C by 10 m. By what time
interval does runner A beat runner C?

Homework Equations


d = 100 m
Δd = 10 m
Δt[itex]_{i}[/itex] = time for runner i to travel 100 m
v[itex]_{i}[/itex] = speed of runner i

The Attempt at a Solution


Can this problem be solved without knowing the speed of one of the runners? Here's a few equalities for this problem:

v[itex]_{a}[/itex] = [itex]\frac{d}{Δt_{a}}[/itex]
v[itex]_{b}[/itex] = [itex]\frac{d-Δd}{Δt_{a}}[/itex]
v[itex]_{c}[/itex] = [itex]\frac{v_{b}Δt_{b}-Δd}{Δt_{b}}[/itex]

Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
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Try assuming a couple of different velocities for runner A and then work out the time difference. Does the time difference vary with the velocity of runner A? This should answer your question.
 
  • #3
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Note that runner A beats runner C by more than 20 m.
 
  • #4
rude man
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Note that runner A beats runner C by more than 20 m.
Yikes, that's right!

I am deleting my post accordingly to think more about this.
 
  • #5
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I tried assuming a few speeds for runner A and found that the interval between runners A and C depends on the speed of runner A. In other words, there is no unique solution with the given information.
 
  • #6
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The problem has six unknowns (3 speeds and 3 times) and five equations. It is fairly trivial to express any five unknowns in terms of the other one.
 
  • #7
rude man
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Right all around. At least I got the 'no unique solution' part right ...
 
  • #8
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time for C

A beats C by a time period of 20/VC

VC=velocity of C
 
  • #9
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A beats C by a time period of 20/VC

VC=velocity of C
This assumes that A beats C by 20 meters. This is not given. What is given is that B beats C by 10 meters, which happens when B is at the finish line. When A is at the finish line, B is not there, so you can't assume that he is 10 meters ahead of C yet.
 

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