Finding the range of a projectile?

Question

A baseball is hit 3 ft. about the ground at 100 ft./sec. and at and angle of 45 degrees with respect to the ground.

I have found:
vector-valued function is
r(t)=(50tsqrt2)i+[3+(50tsqrt2)-16t^2]j
and maximum height= 81 ft
how do I find the range?
and if possible can you help set up the integral for the arc length?

Thank You

ultimate reality1 · Accepted Answer

Using your method, the range would seem to be the integral of that vector. 

Arc length would be 

r = √[x'(t)^2 + y'(t)^2] dt

So that's the derivative of the i vector squared added to the derivative of the j vector squared. Then the square root of all of that stuff and you find the arc length of the curve.

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Physics method, you computed the maximum height, which means you found the time it takes for the ball to reach that height. This means you just multiply the time it takes to reach its high by 2 to find the total air time of the ball. Then you just apply one of the kinematics equations for horizontal velocity. 
range = Vox * total airtime
You can find the Vox from the initial Velocity given and the angle.

Finding the range of a projectile?

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