61 Parachute Jumping,Falling,and Landing David C.Arney,Barbra S.Melendez,Debra Schnelle1Introduction It is extremely important that leaders of airborne units understand the safety,medical,and operational issues involved in parachute jumping.Mathematical modeling of the flight and the landing of either a person or a cargo package can help with this understanding.In this section,we describe,analyze,and model several components of parachute jumping.These are basic models using simplified data in order to begin the study of these issues and should not be used for actual airborne operations.The Army has numerous technical manuals that govern the technical issues associated with airborne operations.The Situation An airborne jumper,whose mass is 103 kg,leaps with a velocity of 0.555 m/s straight out(direction of the x-axis)from the side of an airplane which is moving at a velocity of 115 m/s(direction of the y-axis).The airplane is flying at an altitude(height on the z-axis)of 4000 meters.When the jumper leaves the aircraft,he is high enough in altitude that the air resistance can be considered negligible,and he falls as a freely falling body for 11.5 seconds until he pulls the parachute cord.After that point,the fall becomes a three dimensional projectile motion problem with non-constant acceleration along each axis as air resistance must now be taken into account.The reference axes are established as follows(x-axis in direction of jumper leaving plane,y-axis in direction of plane when the jumper exits,z-axis is vertical,and the origin is on the ground directly under the plane the moment the jumper exits).When the parachute cord is pulled,we assume that the parachute deploys immediately and exerts a combined drag and air resistance force of magnitude D=dv2,where d is the drag coefficient due to the parachute and air resistance,and v is the velocity of the jumper.In this case,we use d=20,but normally the value of d is difficult to determine,since it depends on so many variables.Assume the drag force is exerted only along the vertical direction(z-axis),and that the effect of air resistance along the y-direction can be modeled as a force of magnitude F=-bv,where the minus sign indicates that the force opposes the 1 Formerly in Department of Physics,USMA 名师资料总结-精品资料欢迎下载-名师精心整理-第 1 页，共 9 页 -62direction of motion and b is the drag coefficient due to air resistance.In this case,we use b=10.Simultaneous to the opening of the parachute,the jumper experiences a cross wind in the positive x-direction of 1.2 m/s.The force due to this cross wind can be modeled as G=b(w -v),where b is the drag coefficient due to air resistance(b is set to 10 above),v is the speed of the jumper,and w is the cross wind speed.The collision between the jumper and the ground can also be modeled using mathematics and physics.There are several medical and safety factors that need to be considered for such a collision.Rapid deceleration of the head can be very serious due to the sheer strain on the brain stem(shear means that the shape of an object has changed as the result of forces).A measure of this shear strain to the head is the severity index I,which is determined by the equation I=2(v)/(g?t)2.5(?t),where v is the velocity at impact,?t is the duration of impact,and g is the acceleration due to gravity.When the severity index for a collision is above 1000,the collision is fatal.When the value of the severity index is approximately 400,unconsciousness and mild concussion are the result.The stress on the long bones of the legs during the collision is compressive.When the compression force per unit area exceeds the ultimate tensile strength of the bone(given below for various bones,with their associated cross-sectional areas),the bone breaks in compression.Bone Ultimate Tensile Average Strength Cross-Sectional(N/m2)Area(m2)Femur 1.21 x 1085.81 x 10-4Tibia 1.40 x 1083.23 x 10-4Spinal Cord(back)2.20 x 1084.42 x 10-4Spinal Cord(neck)1.80 x 1084.42 x 10-4 Table 1 名师资料总结-精品资料欢迎下载-名师精心整理-第 2 页，共 9 页 -63The compressive force F experienced as a result of the impact on the earth can be determined from a physics model using momentum.Momentum p is defined as p=mv and the Force model is F=?p/?t,where?p is the change in momentum and?t is the time duration of the collision.If the jumper lands on the ground,we assume that he performs a perfect parachute landing fall,as he was taught in his free fall class.The time of his impact will depend upon how soft the ground is where he lands.Snow or soft sand,for example,would significantly extend his duration of impact and thus reduce the force of impact.More Simplifying Assumptions For our discussion and the initial modeling of this situation,we use the following assumptions in addition to or amplifying those already discussed:?We are not considering the complexities of landing such as landing on uneven ground or in the trees or in an inappropriate manner.?We assume the parachute opens instantaneously and the drag force D