Background

    In calculus, a model for projectile motion with no friction is considered, and a "parabolic trajectory" is obtained.  If the initial velocity is    and    is the initial angle to the horizontal, then the parametric equations for the horizontal and vertical components of the position vector are
    
(1)        ,  
    and
(2)        .  

Solve equation (1) for t and get  ,  then replace this value of t in equation (2) and the result is

         ,

which is an equation of a parabola.  

    The time required to reach the maximum height is found by solving  :  

        ,
yields  
        ,
        
and the maximum height is

        .

    The time till impact is found by solving  , which yields  , and for this model,  .  The range is found by calculating  :  

        .
        
For a fixed initial velocity  , the range    is a function of  ,  and is maximum when  .

Numerical solution of second order D. E.'s

    This module illustrates numerical solutions of a second order differential equation.  First, we consider the special case where the projectile is fired vertically along the y-axis and has no horizontal motion, i. e.  x(t)= 0.  The effect of changing the amount of air drag or air resistance is investigated.  It is known that the drag force acting on an object which moves very slowly through a viscous fluid is directly proportional to the velocity of that object.  However, there are examples, such as Millikan's oil drop experiment, when the drag force is proportional to the square of the velocity.  Further investigations into the situation could involve the Reynolds number.  

Math-Models (Projectile Motion I)  The following mathematical models are are considered.  
(i).     No air resistance  ,   and
        .

(ii).     Air resistance proportional to velocity  ,  and
        .

(iii).     Air resistance proportional to the square of the velocity  ,  for the ascent, and  ,  for the descent, and
        

(iv).     Air resistance proportional to the power of the velocity
        

Proof  Projectile Motion  Projectile Motion  

Computer Programs  Projectile Motion Projectile Motion  

Mathematica Subroutine (Runge-Kutta Method for second order D.E.'s)  To compute a numerical approximation for the solution of the initial value problem  

      with  , ,  

over the interval    at a discrete set of points.  

Example 1.  An arrow is shot upward from the origin with an initial velocity of  300 ft/sec.  Assume that there is no air resistance and use the model  
        .  
Find the velocity and position as a function of time.  Find the ascent time, the descent time, maximum height, and the impact velocity.
Solution 1.

Example 2.  An arrow is shot upward from the origin with an initial velocity of  300 ft/sec.  Assume that air resistance is proportional to the velocity, , and use the model  
        .  
Find the velocity and position as a function of time, and plot the position function.  Find the ascent time, the descent time, maximum height, and the impact velocity.
Solution 2.

Example 3.  An arrow is shot upward from the origin with an initial velocity of  300 ft/sec.  Assume that air resistance is proportional to the square of the velocity,  , and use the model  
        .  
Find the velocity and position as a function of time, and plot the position function.  Find the ascent time, the descent time, maximum height, and the impact velocity.
Solution 3.

Example 4.  An arrow is shot upward from the origin with an initial velocity of  300 ft/sec.
Assume that air resistance is proportional to the power of the velocity,  ,   and use the model  
        .  
Find the velocity and position as a function of time, and plot the position function.
Solution 4.

Trajectory of a baseball.

More Math-Models (Projectile Motion II)  The following mathematical models are are considered.  
(v).     No air resistance  
          

(vi).     Air resistance proportional to velocity
          

(vii).     Air resistance proportional to the square of the velocity
        

Proof  Projectile Motion  Projectile Motion  

Mathematica Subroutine (Runge-Kutta Method for two second order D.E.'s)  To compute a numerical approximation for the solution of the initial value problem  
      with  , ,  
and
      with  , ,  over the interval    at a discrete set of points.  

Example 5.  Consider the frictionless model for the path of a baseball.  
          
Check out this mathematical model for several cases.  
Solution 5.

Example 6.  Consider the linear drag model for the path of a baseball.  
          
Check out this mathematical model for several cases.  
Solution 6.

Example 7.  Consider the quadratic drag model for the path of a baseball.  
        
Check out this mathematical model for several cases.
Solution 7.

Example 8.  The model in Example 3 is the quadratic drag model.  The following is not.
          
Let's check it out.
Solution 8.

Research Experience for Undergraduates

Projectile Motion  Projectile Motion   Internet hyperlinks to web sites and a bibliography of articles.  

Download this Mathematica Notebook Projectile Motion

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(c) John H. Mathews 2004