Projectile Motion

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Projectile Motion Lesson for Calculus Students

 A projectile motion problem involves a two dimensional analysis. Initial velocity is broken into horizontal and vertical components. Visit the following link to view a graphic of the situation: http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html#tra12

 Use trigonometric functions to break down velocity into its horizontal and vertical components.

 Horizontal component v_{xo = Vo}cos A

Vertical component     v_{yo = Vo}sin A       

 

Where A is the angle in degrees in which the object in thrown.

 

 

A general kinematic equation derived earlier in the semester was:

 

y = y_o + v_yot - ½ at²                        Assuming acceleration is acting downward due to gravity.

 

where   y   =  horizontal distance

            y_o = initial distance

            v_o = initial velocity

            t = time in the air

            a = acceleration due to gravity (9.8 m/s² on Earth)

 

In calculus it is studied that taking the derivative of a distance equation provides a new equation for velocity.

 

Taking the derivative of this equation results to an equation for final velocity: v_y = v_yo - at.

 

Since the base equation is quadratic form, its graph is parabolic, thus describing the path of a projectile problem. In calculus it is noted that maximums and minimums in the original function are zeros in the derivative. Setting the derivative equal to zero will allow to solve for time for maximum height reached.

 

0 = v_yo  - at     rewrite to

_{Vyo =} at            then

t =  _Vyo/a 

 

Assuming a reverse free fall problem, maximum height reached can be determined using y_max= 1/2at² .

 

To determine total flight time, this time is doubled assuming rise time is equal to downward time t_{t =} 2t .

 

Neglecting air friction, therefore assuming constant velocity, horizontal range is calculated:

R = v_xot_t

 

Summary of Procedures to Calculate Projectile Motion Problems.

 

1)      Determine horizontal and vertical components of initial velocity.

2)      Determine time to reach maximum height: t = v_y/a

      3) Determine maximum height: y = ½ at²

      4) Determine total time in air:  t_t = 2t

      5) Determine total range covered: R = v_xot_t

 

Assessment

 

1)      a. Use the following link to determine answers to the following problem. Set initial conditions and run the simulator: http://lectureonline.cl.msu.edu/~mmp/kap3/cd060.htm

            Problem: A cannon ball is launched at an angle of 30 degrees with an initial velocity of 9   

            m/s. Determine maximum height reached and range covered.

            b. Model the situation with a base equation: y = y_o + v_yot - ½ at²

                c. Calculate time for maximum height achieved by taking the derivative.

d. Email complete solutions to this problem to the instructor discussing comparisons found with using the simulator and also including total time.

      2) The path of a projectile thrown object is modeled through an equation y = 7.07t –4.9t²

                Use the derivative to determine flight time and initial conditions. Also calculate maximum    

           height and range reached. Use the link provided to model the situation. Compare and     

          discuss results. Email all solutions and discussion to the instructor.

3) Use the link http://lectureonline.cl.msu.edu/~mmp/kap3/cd060.htm to set determine initial conditions and use the base equation, y = y_o + v_yot - ½ at² , to model the situation. Design a word problem with stating he equation. Through email, trade word problems with classmates to determine maximum height and range, and total flight time. Submit all final calculations to the instructor through email.