You are the technical advisor for the David Letterman Show Y

You are the technical advisor for the David Letterman Show. Your task is to design a circus stunt in which Super Dave, who weighs 750 N, is shot out of a cannon that is oriented 40 degree above the horizontal. The \"cannon\" is actually a 1 m diameter tube that uses a stiff spring to launch Super Dave. The manual for the cannon states that the spring constant is 1800 N/m. The spring is compressed by a motor until its free end is level with the bottom of the cannon tube, which is 1.5 m above the ground. A small seat of negligible mass is attached to the free end of the spring for Dave to sit on. When the spring is released, it extends 2.75 m up the tube. The seat does not touch the sides of the 3.5 m long tube, so there is no friction. After a drum roll, the spring is released and Super Dave will fly through the air. You have an airbag 1 m thick for Super Dave to land on. The airbag will exert an average retarding force of 3000 N in all directions. Assume that this force is not enough to harm Super Dave. Note that this retarding force does not include the force due to gravity. You need to determine if the airbag is thick enough to stop Super Dave safely (with a good safety margin) - that is, he comes to a stop before he reaches ground level. Consider the seat and spring to have negligible mass. Ignore air resistance. Note that this is a real life problem and may have extraneous information. For this entire problem, we will consider the reference height, h = 0, to be ground level. Assume that Dave lands perpendicular to the airbag\'s surface for a good safety margin.

Solution

The spring constant of the cannon = 1800 N/m The spring extension = 2.75 m Hence, the force on the man, Dave, by the spring= 1800 X 2.75 N = 4950 N Now Mass of Dave = 750 N/g = 75 Kg assuming acceleration due to gravity g - 10 m/s^2 So the initial acceleration = 4950 N/ 75 Kg = 66 m/s^2 The horizontal component of this acceleration= 66 X cos 40 m/s^2 = 66 X .77604 m/s^2 = 50.55 m/s^2 To calculate the initial velocity of Dave ,we have to consider the spring\'s motion. If a spring of spring constant k is compresses by a mass m, its frequency w^2 = k/m = 1800/75 N/m/Kg = 24 /s^2 and velocity v= w {\\sqrt(A^2 - x*2)} If t is the time taken for the spring to stretch, then v = at= 66t but t = 1\\ w = 1/ sqrt{24| s so we get v = 66/ sqrt{24| s This is the inital velocity. Now the maximum range will be = v^2 /g = [( 66 X 66) /24] /10 substituting the value of v & assuming g = 10 m/s*2 This gives R = 18.25m But there is an initial x-displacement due to the size of the apparatus. This can be calculated from geometry: tan 40 = 1.5m/ x_0 Dave will reach the maximum height when his weight will balance t he upward thrust and net velocity in y direction will be zero. Then he will start descending. Now due to symmetry of the problem, his downward force will be F= weight (mg) + force of descent(ma) Typically, this force can be calculated by using the following facts: The intial x -displacement : x_0 = 1.5 m / tan40=1.69 m The vertical component of the initial force acting on Dave = 4950 X sin 40 N = 4950 X 0.6428 N =3181N now, this force acts upwards while the weight of Dave act downward. So the net upward force on Dave= (3181 - 750)N = 2431 N But for safety reason, the maximum value of the downward force can be as high as 4950N as air resistance is negligible. So IT IS NOT SAFE to have this circus.