RANGE MAXIMIZATION UNDER AIR RESISTANCE

GOAL Determine the angle of elevation of a gun which maximises the reach of a gun, assuming gravity and air resistance are the only forces acting on the shell.

PROBLEM A gun fires shells of mass 1 kg at a speed of p m/sec . The gun is fired at an angle b to the horizontal. We assume that the region is flat, b is in radians and . During its flight the shell experiences

• a gravitational pull (g = 10 )
• air resistance, of magnitude 1/3 the speed of the shell, acting in a direction opposite the velocity of the shell (actually air resistance depends on the shape of the shell, density of the air, etc.).

1. By hand, determine the position of the shell at time t? Then, using Maple, verify that your solution is the solution of the differential equation with the correct initial position and initial velocity.
2. Assuming p=500, where does the shell land when ?
3. If p=500, at what angles (in degrees) may the shell be fired if it is to land on the ground 1200 metres away from the gun.
4. Assuming p=500, what angle b (in degrees) makes the shell land farthest from the gun. What is the range of this gun?

REMARK Note that in question 2, , but not so in questions 3 and 4. So when using Maple for questions 3 and 4, be sure to unassign the value of b so that it does not assume that . This may be done by

b := 'b' ;

 
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