Projectile Motion

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Projectile Motion Lesson for Algebra 2 and Pre-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

 Vertical analysis

The upward motion of an object can be calculated as a reverse free falling problem.

First velocity must be broken into its vertical component using the sin ratio.

 

v_y =  v_osinA

 

where         v_{y =} velocity in the vertical direction.

                  v_o = initial velocity of the object thrown

                  A = measure of the angle in degrees of the object thrown

 

When an object reaches its maximum height final velocity is zero. Using a previous equation v = v_o + at can be used to calculate time.

 

When v = 0, the equation becomes 0 = v_{o –} at  (note: a is negative because gravity is acting downward. Manipulating this equation in terms of vertical motion gives: v_{y =} at

 

Time to reach maximum height can therefore be calculated through: t = v_y/a.

Since the object will also fall at the same amount of time, time is doubled to calculate total time (t_t) in air: t_t = 2t

 

      Also, since it takes the same amount of time for an object to drop as it did when it rises, maximum height can be calculated assuming a free fall problem and using a previous equation: y = ½ at²

 

Horizontal Analysis

 

Velocity must be broken down into its horizontal component (v_x) using the cos function.

 v_{x = Vo}cos A

 Neglecting air friction, Range covered by the object can be calculated by using a previous equation: R = v_xt_t

 

Summary of procedures for a projectile motion problem.

 

Vertical Motion

1)      Determine the vertical component of initial velocity: v_y =  v_osinA

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

 

      Horizontal Motion

1)      Determine the horizontal component of initial velocity: v_{x = Vo}cos A

2)      Determine range covered: R = v_xt_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. Email complete solutions to this problem to the instructor discussing comparisons found with using the simulator and also including total time.

2)      Using the simulator, design a word problem of the situation setting initial conditions. Trade problems through Email with a classmate to solve. Email all solved problems to the instructor.

3)      Email a complete solution to the following problem to the instructor. An arrow is shot at   65 degrees with an initial velocity of 20 m/s. Determine maximum height, range, and total time in the air. Use the following link to check and compare answers: http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html#tra12