How Do You Calculate the Arc Length of a Baseball's Trajectory?

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Discussion Overview

The discussion revolves around calculating the arc length of a baseball's trajectory based on its initial conditions, including height, distance, and launch angle. Participants explore the use of parametric equations to model the ball's path and seek assistance in determining the arc length until the ball lands.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Exploratory

Main Points Raised

  • Participants present parametric equations for the ball's trajectory based on initial conditions, with one participant stating x = 146.7cos(theta)t and y = 3 + 146.7sin(theta)t - 16t^2.
  • There is a query about whether the ball reaches the centerfield fence, with a participant concluding it does not, as the ball only travels 349 ft.
  • One participant provides a formula for arc length, expressing it as an integral involving the derivatives of the parametric equations.
  • Participants express uncertainty about the limits of integration for the arc length calculation, with one suggesting the use of the time at which the ball lands, denoted as T.
  • There is a discussion about determining T, with suggestions to use an angle of 15 degrees or to keep it as an arbitrary constant theta.

Areas of Agreement / Disagreement

Participants generally agree on the need to determine the time of flight T for the arc length calculation, but there is uncertainty regarding the appropriate angle to use, leading to multiple viewpoints on how to proceed.

Contextual Notes

Limitations include the dependence on the angle theta, which remains unspecified, and the unresolved nature of the integration limits for the arc length calculation.

calcboi
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The centerfield fence at a ballpark is 10 ft high and 400 ft from home plate. The ball is 3 ft above the ground when hit, and leaves with an angle theta degrees with the horizontal. The bat speed is 100 mph. Use the parametric equations x = (v0cos(theta))t y = h + (v0sin(theta))t - 16t^2
a. Write parametric equations for the path of the ball.
I got this as x = 146.7cos(theta)t and y = 3 + 146.7sin(theta)t - 16t^2
b. Graph the path of the ball if theta = 15 degrees. Is it a home run?
I got no because the ball only reached 349 ft, not 400 ft.
c. Find the arc length of the path of the ball until it lands.
This is the part I need help on. I don't know how to get the arc length.
 
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calcboi said:
The centerfield fence at a ballpark is 10 ft high and 400 ft from home plate. The ball is 3 ft above the ground when hit, and leaves with an angle theta degrees with the horizontal. The bat speed is 100 mph. Use the parametric equations x = (v0cos(theta))t y = h + (v0sin(theta))t - 16t^2
a. Write parametric equations for the path of the ball.
I got this as x = 146.7cos(theta)t and y = 3 + 146.7sin(theta)t - 16t^2
b. Graph the path of the ball if theta = 15 degrees. Is it a home run?
I got no because the ball only reached 349 ft, not 400 ft.
c. Find the arc length of the path of the ball until it lands.
This is the part I need help on. I don't know how to get the arc length.

Hi calcboi!

You can find the formula for arc length here on wiki.

In short the arclength L is:
$$L=\int ds = \int \sqrt{dx^2 + dy^2} = \int \sqrt{(x'(t)dt)^2 + (y'(t)dt)^2} = \int \sqrt{x'(t)^2 + y'(t)^2}dt$$
 
Thanks, but I am unsure what to use for the limits of integration.
 
calcboi said:
Thanks, but I am unsure what to use for the limits of integration.

Did you find the time at which the ball lands?
Let's call that T.
Then the integral limits are t=0 and t=T.
 
I like Serena said:
Did you find the time at which the ball lands?
Let's call that T.
Then the integral limits are t=0 and t=T.

I can't find T unless I have an angle for theta, and I don't know what to put for the angle. I am guessing I should use 15 degrees from the earlier portion but I'm not sure.
 
calcboi said:
I can't find T unless I have an angle for theta, and I don't know what to put for the angle. I am guessing I should use 15 degrees from the earlier portion but I'm not sure.

I'm also guessing you should probably use 15 degrees from the earlier question.
But you can also leave it as just theta and treat it as an arbitrary constant which just happens to have an unknown value.
 

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