Question

Emmanuel Zacchini launched as a human cannonball in 1940 soared over three Ferris wheels each 18.0 m high covering a horizontal distance of 10.63m. a.) Assuming that the point of projection is 2.50 m above the ground and that he landed safely on a net placed at the same level, find his initial velocity. b.) Assuming that he missed the net and landed on the ground, with what velocity will he strike the ground?

Answer 1

We have range and height which can give us the tangent of the angle of launch. Note: the height we're going to use here is the height between the highest point of trajectory and the 2.5 m where the launch was initiated:

$$H=\frac{v^2\sin^2\theta}{2g}=\frac{v^2·\sin\theta·\sin\theta}{2g},\\\space\\ R=\frac{v^2\sin(2\theta)}{g}=\frac{v^2·2\sin\theta·\cos\theta}{g},\\\space\\ \frac HR=\frac14\tan\theta,\\\space\\ \theta=\arctan\frac{4H}{R}=\arctan\frac{4(18-2.5)}{10.63}=80°.\\\space\\ v=\sqrt\frac{Rg}{\sin(2\theta)}=17.5\text{ m/s}.$$

Assuming that he missed the net and landed on the ground:

$$u_y=\sqrt{2g(H+h)}=18.8\text{ m/s},\\ u_x=v_x=v\cos\theta=3.04\text{ m/s},\\ u=\sqrt{u_x^2+u_y^2}=19\text{ m/s}.$$