Question

erson standing at the top of a hemispherical rock of radius R kicks a ball (initially at rest on the top of the rock) to give it horizontal velocity V⃗i as shown in above figure. (a) What must be its minimum initial speed if the ball is never to hit the rock after it is kicked? (b) With this initial speed, how far from the base of the rock does the ball hit the ground?

Answer 1

$y_b=R-\frac{gt^2}{2},$

$t=\frac xv,$

$y_b=R-\frac{g}{2v^2}\cdot x^2,$

$t=\sqrt{\frac{2R}{g}},$

$v\geq \frac Rt=\sqrt{\frac{gR}{2}},$

$y_r^2+x^2=R^2,$

$y_b\geq y_r,$

for $x=x_{min}~~$ $y_b=y_r,$

$R-\frac{gx^2}{2v^2}=\sqrt{R^2-x^2},$

$R^2-\frac{gRx^2}{v^2}+\frac{g^2x^4}{4v^4}=R^2-x^2,$

$1-\frac{gR}{v^2}+\frac{g^2x^2}{4v^4}=0,$

$\frac{g^2x^2}{4v^2}=gR-v^2,$

$x_{min}=x=\frac{2v}{g}\sqrt{gR-v^2}=|v=\sqrt{\frac{gR}{2}}|=R,$

$l=x_{min}-R=R-R=0.$