A projectile is fired from the origin O with velocity vo and at an angle a to the horizontal through a resistive atmosph

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A Projectile Is Fired From The Origin O With Velocity Vo And At An Angle A To The Horizontal Through A Resistive Atmosph 1 (41.87 KiB) Viewed 43 times

A Projectile Is Fired From The Origin O With Velocity Vo And At An Angle A To The Horizontal Through A Resistive Atmosph 2 (32.23 KiB) Viewed 43 times

A projectile is fired from the origin O with velocity vo and at an angle a to the horizontal through a resistive atmosphere, to land on the same horizontal plane some distance away. During its flight the projectile experiences an air resistance that is proportional to the projectile's velocity, but which diminishes exponentially with height. That is, the projectile experiences a resistance force which follows the empirical law F(v, 2) = - [be-/h] v₁ where b is a (constant) resistance coefficient, h is a (constant) length scale, and v is the projectile's velocity at time t. The path taken by the projectile is given by the solution of the dynamic equation based on Newton's second law of motion: mr(t) = -mgj + F(r. z). It has been found that a projectile passing through a so-called "thinning" atmosphere (such as one represented by Eq (1)) achieves a longer range than is predicted by the exact result of a projectile's path through a uniform atmosphere with an air resistance coefficient of b, under otherwise identical conditions.
(A) Establish an appropriate coordinate system to represent the dynamics. (B) Describe the respective components of the motion, reflecting on any interdependencies and their consequences. (C) Produce numerical solutions of the parametric equations of motion for the projectile's trajectory, assuming suit- able numerical values of the initial conditions. (D) If possible, derive a closed form expression for the solution of the vertical component of the equation of motion, and compare this expression with your numerical solution determined in Part (C). (E) Determine the angle of projection a that would result in the maximum range of the projectile and determine the latter, assuming the same initial speed. Compare these with the corresponding angle and range obtained for a projectile passing through a uniform atmosphere.