Projectile Equations (2-Dimensional Motion) ignoring air resistance

(also known as "The Magnificent Seven")

(try an internet search of "Projectile Motion Java" for good animations of projectile motion)

The basic problem for projectile motion is an object launched at an angle while gravity is pulling on it. An angle and an initial velocity are usually given to you THAT ARE ONLY GOOD FOR FINDING THE INITIAL VERTICAL VELOCITY (v_iy) AND THE HORIZONTAL VELOCITY (V_X)!

The format of the problems is usually:

V

q

That "V" is NOT a V_x or a V_y, IT IS A COMBINATION OF THE TWO! So you're first step is to break that V into a V_x and a Viy by using the Sine and Cosine functions:

V Viy = V*sin q q

Vx = V*cos q

Now you have Vx & Viy and can forget about the 'V' and the q.

For the Vertical parts (components):

d_Y = v_iYt + ½ gt²

v_fy²= v_iy² + 2gd_Y

v_fy = v_iy + gt

Use these equations to find:

How high (d_y)

Final vertical velocity (v_fy)

Initial vertical velocity (v_iy)

For the Horizontal parts (components):

d_x = v_X*t

Use these equations to find:

How far (d_x)

Note: TIME IS THE ONLY VARIABLE THAT CAN BE USED ON BOTH SETS OF EQUATIONS

Points of Confusion for students:

1. Students try to use the "V" from the very beginning of the problem in the equations…Don't! It is neither a vertical (v_iy, v_fy) nor a horizontal velocity (v_X)!
2. Students get d_x and d_y mixed up or they don't understand why there are 2 distances- d_y tells you the HOW HIGH the object goes and d_x tells you HOW FAR AWAY IT LANDS!
3. They don't know why v_fy is sometimes zero and sometimes it's the same as v_iy (but has the opposite sign).
4. v_fy is zero when you need to know how HIGH the object goes because it stops at the top!

   BUT,

5. v_fy is equal to v_iy when the object comes crashing back to the ground! (objects come down at the same speed they went up!)

   4. They don't know why they have two different times. One time will be how long

   it takes the object to make it to the top of its arc (here v_fy is zero), and the other time is how long the object is in the air (here, v_fy is equal to v_iy). By the way, the time up should be one-half of the total time so if you solve for the wrong time, you can just half or double your answer to correct.

   Notes:

   The horizontal velocity (v_X) NEVER CHANGES over the entire trajectory- this makes it a CONSTANT.

   The vertical velocities (c, c) NEVER STAY THE SAME because gravity eats away (subtracts) from the v_iy all the way up and adds to the v_fy all the way down!

   t_up = t_down

   v_X Always the same

   v_{fyv =} 0 (the top)

   v_{iy = -} v_fy v_{iy = -} v_fy