Calculate the ratio of the time the jumper is above ymax/2

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In summary: Good job!In summary, the conversation discusses an athlete jumping vertically and the calculation of the ratio of the time the athlete is above Ymax/2 to the time it takes to reach Ymax from the floor. The conversation delves into the use of kinematics and gravitational potential energy to calculate the velocity at Ymax/2 and ultimately simplifying the equation to a ratio of 2 to sqrt2 - 1.
  • #1
killerinstinct
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An athlete jumps vertically. treat athlete as particle and Ymax is the maximum height above the floor the athlete achieves. To explain why he seems to hang in the air, calculate the ratio of the time he is above ymax/2 to the time it takes him to go from the floor to that height.

we get 2 equations,
Vymax/2 = V0Y-gt_1
Vymax = Vymax/2 - gt_2

after isolating t
we get:
2t_2 / t_1 = 2 Vymax/2 / V0y-Vymax/2

V0Y - Vymax/2 < Vymax/2
<<<< how do you go about explaining this in common language??

the time it takes for person to reach ymax/2 from Yground is les thatn time it takes the person to reach Ymax from Ymax/2, so Vavg from Yground to Ymax/2 is greater thatn Ymax/2 to Ymax.
 
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  • #2
You already know that Vymax = 0

Use gravitational potential energy to calculate the velocity at ymax/2 in terms of v0

Then you'll know the velocity at the bottom... v0, velocity at ymax/2, and the velocity at ymax which equals 0.

so then you can calculate the times... why did you 2t_2 / t_1 instead of t_2/t_1?
 
  • #3
i set t_2 as time from ymax/2 to ymax. so the time above ymax/2 is 2t2.
gravitational potential??
mgh = mv^2/2??
 
  • #4
on this problem, I'm only allowed to use kinematics. as much as i want to use energy, i can't.
 
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  • #5
killerinstinct said:
i set t_2 as time from ymax/2 to ymax. so the time above ymax/2 is 2t2.
gravitational potential??
mgh = mv^2/2??

EDIT: Never mind you're absolutely right about the 2t_2/t_1. I misunderstood the question.

Yeah, mgh = mv^2/2 was what I was thinking.

You can also get the velocity at ymax/2 using...

vf^2 = vi^2 + 2as
 
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  • #6
ya but t1 is time from bottom to middle. the question asks for 2t2/t1 not t2/t1. but anyways, that's technical... no big deal.

wouldn't using vf^2 = vi^2 +2as make this more complicated than it is??

i already got a ratio of 2Vymax/2 to Voy-Vymax/2
there is a way to simplify that?? by substituting in another equation. I don't see how it would simplify it. maybe I'm just bad at algebra, can you show me?
 
  • #7
killerinstinct said:
ya but t1 is time from bottom to middle. the question asks for 2t2/t1 not t2/t1. but anyways, that's technical... no big deal.

Yeah, you're right. I was misunderstanding.


wouldn't using vf^2 = vi^2 +2as make this more complicated than it is??

i already got a ratio of 2Vymax/2 to Voy-Vymax/2
there is a way to simplify that?? by substituting in another equation. I don't see how it would simplify it. maybe I'm just bad at algebra, can you show me?

2t_2 / t_1 = 2 Vymax/2 / V0y-Vymax/2

This doesn't simplify unless you write Vymax/2 in terms of V0y... the question wants you to get the actual number for this ratio...

Also be careful not to confuse Vymax/2 with (Vymax)/2 = 0 (since Vymax = 0)... might be better to write Vx where x = ymax/2... or something like that.
 
  • #8
vx^2=v0y^2 + 2 a ymax/2

you get:
2 sqrt (Voy^2-2gYmax/2) to Voy- sqrt (V0y^2-2gymax/2) RATIO

how do you simplify that??
 
  • #9
killerinstinct said:
vx^2=v0y^2 + 2 a ymax/2

you get:
2 sqrt (Voy^2-2gYmax/2) to Voy- sqrt (V0y^2-2gymax/2) RATIO

how do you simplify that??

Use the same equation again, but this time to get ymax in terms of v0. ie: vx = 0, s = ymax.
 
  • #10
Ymax = Voy^2/2g.
sub in and i get
2sqrt(2gYmax/2) to V0y-sqrt(2gYmax/2)

sub in V0y

and i get
2sqrt(2gYmax/2) to sqrt (2gYmax) - sqrt (2gYmax/2)

some algebra help... how you simplify what's in bold...

if I'm doing it right... i get 2 to 1 ratio?? that doesn't sound right.
 
  • #11
killerinstinct said:
Ymax = Voy^2/2g.
sub in and i get
2sqrt(2gYmax/2) to V0y-sqrt(2gYmax/2)

sub in V0y

and i get
2sqrt(2gYmax/2) to sqrt (2gYmax) - sqrt (2gYmax/2)

some algebra help... how you simplify what's in bold...

if I'm doing it right... i get 2 to 1 ratio?? that doesn't sound right.

sqrt(gYmax) cancels from the numerator and denominator...

You should get: [tex]\frac{2}{\sqrt{2} - 1}[/tex]
 
  • #12
wait... i did it again and got ratio of
2 to sqrt2 - 1
is that right??
 
  • #13
i guess i did it right..
i knew the physics of this problem. but i couldn't figure out the algebra.
greatly appreciate your algebra lesson. thank you very much.
 
  • #14
killerinstinct said:
i guess i did it right..
i knew the physics of this problem. but i couldn't figure out the algebra.
greatly appreciate your algebra lesson. thank you very much.

no prob! good job!
 
  • #15
killerinstinct said:
wait... i did it again and got ratio of
2 to sqrt2 - 1
is that right??

yeah, that's what I got.
 

What does "Calculate the ratio of the time the jumper is above ymax/2" mean?

This phrase refers to finding the proportion of time that a jumper spends above half of their maximum height (ymax).

Why is this ratio important to calculate?

Calculating this ratio can provide insight into the jumper's performance and technique. It can also be used to compare with other jumpers or to track progress over time.

How do I calculate this ratio?

To calculate this ratio, you will need to know the total time the jumper was in the air and the time they spent above half of their maximum height. Divide the latter by the former and multiply by 100 to get the percentage.

What factors can affect this ratio?

The ratio can be affected by the jumper's technique, physical abilities, and any external factors such as wind or equipment. It can also vary depending on the type of jump being performed.

Can this ratio be used in other sports or activities?

Yes, this ratio can be applied to any activity that involves jumping or reaching a maximum height, such as high jump, basketball, or even gymnastics. It can also be adapted to measure other variables, such as the ratio of time spent in a certain position or performing a specific movement.

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