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

[calcboi]

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.

 

[I like Serena]

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:
$$L=\int ds = \int \sqrt{dx^2 + dy^2} = \int \sqrt{(x'(t)dt)^2 + (y'(t)dt)^2} = \int \sqrt{x'(t)^2 + y'(t)^2}dt$$

 

[calcboi]

Thanks, but I am unsure what to use for the limits of integration.

 

[I like Serena]

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.

 

[calcboi]

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.

 

[I like Serena]

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.