Question

"You are hired by a carnival to develop a new human cannon launching device. Instead of propelling the human with explosive gunpowder, you are to launch the person with a stretched spring. Upon release, the person is to land in a net a certain distance from the cannon. To test the idea, you use a spring that is slide down around a dowel until the last turn of the spring, which has been bent across the center opening, prevents the spring from sliding down further. You decide to launch the spring by stretching it so that when released it flies from the dowel, through the air, and lands in a box across the room. Develop the theory needed to make this launching system work so that the spring lands in the box on the first shot."

Answer 1

Trajectory man Chair Spring Net a

The idea is we utilize the spring force to project a man in a direction that makes some angle $\alpha$ with horizontal direction. Then man flies in a parabolic trajectory due to projectile motion and falling on a net placed at a distance from the 'Gun' .

Let us place the spring in a tube or barrel of big diameter so that a person can sit in a chair that is connected to one end of spring kept inside a barrel. The other end of spring is firmly fixed with barrel . There is a small slit is provided in the barrel so that through the slit we are able to stretch the spring.

When spring is stretched and released, the spring oscillates, getting compressed and again getting stretched due to energy stored in the spring when it is compressed. At that time force acting on the man by spring is kx , where k is spring constant and x is compressed length . This force pushing the chair and man upward. At this instant , if spring force kx exceeds the weight of the man, i.e., downward component of gravitational force, then the man will be detached from spring and he will be projected with velocity v that satisfies the relation

(1/2)mv² = (1/2)kx²

where (1/2)mv² is the kinetic energy of projected man and (1/2)kx² is energy transferred from spring to man.

The tube or barrel is kept at angle with horizontal , so that projected man's velocity direction makes some angle with horizontal and he is subjected to projectile motion and follows a parabolic path.

Horizontal distance d from point of projection to the net is calculated from velocity of projection and angle of projection. As per this calculation the net is placed so that the man will fall on the net.