Minimum Velocity only given distance

[tripletrouble]

Minimum Velocity only given distance!

Homework Statement

A soccer ball is kicked by a goalie to a position that is 65 meters down field. What is the minimum velocity necessary to achieve this feat?

Homework Equations

v2=vf2+2a(x-xo)

The Attempt at a Solution

v2=02+2a(65m)

 

[edziura]

Assuming no air resistance, the velocity required will be a minimum what the launch angle is 45 degrees above the horizontal. Use the range equation.

 

[tripletrouble]

But no degree angle is specified, does it still have to be 45 degrees? you can still kick a ball and have it roll horizontally

 

[edziura]

True; I thik the statement of the problem implies that the ball is kicked at some angle above the horzontal and when it returns to the ground, it has convered a horizontal distance of 65m.

 

[tripletrouble]

But to use an angle you need an initial velocity, no? None is specified. My professor said that this problem was very difficult and had a trick to it. There's a part b) that says "if it was kicked at 50 degrees instead" where would the ball land?
that might be the clue, so perhaps there's an implied angle?

 

[edziura]

Well, how about this. The range equation is

R = $$\frac{V_{0}^{2} sin 2\theta}{g}$$

If you solve the equation for Vo and differentiate the result with respect to theta (dVo / dtheta), set it equal to zero and solve for theta, you get 45 degrees. That proves that the min speed for a given range occurs when theta is 45 degrees.

If you then solve the range equation for Vo and substitute 45 degrees,

($$V_{0}$$) min = $$\sqrt{R g}$$