Question

The least velocity at which a ball can
be thrown through a distance 10 m is
(g = 10 m s-2)​

Answer 1

Answer:

First, the boy would need breathing apparatus, because to get the true minimum he would need to be in a vacuum - but I’m ignoring that, in the same way as I am ignoring the curvature of the earth; like David Shaffer I am also assuming that the boy is buried up to his armpits in the ground.

Oh, and I’m stealing David Shaffer’s notation.

I agree with David Shaffer that the best angle is in fact 45 degrees, but actually you don’t need to assume it - if you follow David Shaffer’s argument with angle of projection as θ - I don’t know how to do equations properly on Quora but you get very easily

v t cos θ = s

v t sin θ = ½ g t²

applying sin ² + cos ² =1 you get

¼ g² (t²)²-v² t²+ s²=0

Now the minimum will be when this quadratic in t² has coincident roots: the usual condition for this gives (ignoring the ± which is just a coordinate flip)

v² = g s