Speed distance time, 1500mtr race

[stuwalshe]

Hello, Just after a bit of help with the following, I have a solution but don't think it is correct due to me not including the lap size.

Runner A runs at 6.25m/s per sec and runner B at 4.39m/s per sec in a 1500 mtr race.

a, calculate te distance at which runner A laps runner B.
B, if runner B has to finish the race in 5 mins or under calculate how fast he needs to run from the moment he is lapped by runner A, assume 400m laps..

This is what I have done.
a,
time taken for runner A =1500m/6.25 = 240 seconds
time taken for runner B =1500m/4.39 = 341.7 seconds

difference in speed = 341.7/240=1.42

1500m/1.42 = 1056.34 meters.

i.e Runner A laps runner B at 1056.34m

b,
Distance remaining = 1500-1056.34 = 443.66m
time taken so far = 1056.36/4.39m/s = 240.62 seconds

5 mins = 300 secs

Time remaining to finish race = 300-240.62 = 59.30s

distance remaining/time remaining = 443.66/59.30 = 7.47m/s

i.e runner B needs to run at 7.47m/s to complete the race in time.

any feedback would be great
thankyou

 

[Doc Al]

This is what I have done.
a,
time taken for runner A =1500m/6.25 = 240 seconds
time taken for runner B =1500m/4.39 = 341.7 seconds

difference in speed = 341.7/240=1.42

Seems to me you are finding the ratio of the speeds, not the difference. Better would be to find the relative speed, which is the actual difference of speeds.

1500m/1.42 = 1056.34 meters.

i.e Runner A laps runner B at 1056.34m

Do a sanity check. When does runner B get to that point? Where is runner A at that moment? Is that exactly one lap ahead?

You'll find that your answer isn't quite right. Instead, consider the relative speed and figure out at what time runner A gets a complete lap ahead of runner B. (Yes, you must consider the lap distance.)

 

[stuwalshe]

thankyou.

I see what you mean.

I have now taken the difference between the speeds

6.25-4.39 = 1.86m/s

I have also worked out that runner a takes 64sec per lap, (240s/3.75)
runner b takes 91.12sec per lap. (341.7/3.75)

there is 101.7 seconds difference between the two runners finishing the 1500 meters.

if i do similar to my first attempt I would have 1500m/1.86 = 806.45, this does not look correct.
I have also tried to add the two velocities, i.e (6.25+4.39)/2 to give average speed = 5.32.

I am unsure which is correct and also unsure how to figure out when the runner is 400m ahead.

thanks

 

[Doc Al]

I have now taken the difference between the speeds

6.25-4.39 = 1.86m/s

This is the relative speed of the two runners. In other words, from the viewpoint of the slower runner, the faster runner is moving at this speed. So how long does it take the faster runner to gain a lap relative to the slower runner?

if i do similar to my first attempt I would have 1500m/1.86 = 806.45, this does not look correct.

Why use 1500m? That's the length of the entire race, not just a lap!

 

[stuwalshe]

ok,

so the time taken from the point of relative speed for 1 lap is
400m/1.86m/s = 215.05s

215seconds x 4.39m/s = 944m.

215seconds x 6.25m/s = 1344m.

a difference of 400.

thankyou for your help, seams I was trying to make it more complicated than it should be, this then threw me off track, assuming what I have just done is correct.

stu

 

[stuwalshe]

forgot to put the answer,

the distance that runner A laps runner B is 1344m

 

[stuwalshe]

thus the second part of the question changes to

Distance remaining = 1500-944 = 556m
time taken so far = 944/4.39m/s = 215 seconds

5 mins = 300 secs

Time remaining to finish race = 300-215 = 85s

distance remaining/time remaining = 556/85 = 6.54m/s

i.e runner B needs to run at 6.54m/s to complete the race in time.

 

[Doc Al]

Looks good to me!

For part a, if the relative speed idea seemed a bit abstract, you can always just calculate things directly. Write expressions for the position of each runner as a function of time. Then solve for the time when runner A is exactly one lap further than runner B. (Of course, this is equivalent to using relative velocity.)

 

[stuwalshe]

thankyou so much for your help

stu