Question

While playing at Azadi Stadium, Ali threw a soccer ball that reached a maximum height of 30 meters. If he did not defend, he would have landed at a distance of 70 meters. Yes, Ali throws the ball when the ball is 2 meters above the ground and there is no air resistance, at what horizontal distance does the defense catch the ball?

Answer 1

distance:

$x=v_{0x}t$

height:

$h=v_{0y}t-gt^2/2$

maximum height:

$h'(t)=v_{0y}-gt=0$

$h_{max}=h_0+v_{0y}^2/g-v_{0y}^2/2g=30$ m

$v_{0y}=\sqrt{2gh_{max}}=\sqrt{2\cdot 9.8\cdot30}=24.25$ m/s

time of maximum height:

$t_{m}=v_{0y}/g=24.25/9.8=2.47$ s

distance:

$x=v_{0x}t=2v_{0x}t_m=70$ m

$v_{0x}=x/(2t_m)=70/(2\cdot 2.47)=14.17$ m/s

if Ali throws the ball when the ball is 2 meters above the ground:

$h=2+v_{0y}t-gt^2/2$

for h=0:

$4.9t^2-24.25t-2=0$

$t=\frac{24.25+ \sqrt{24.25^2+8\cdot4.9}}{9.8}=5.03$ s

So:

$x=14.17\cdot5.03=71.28$ m