Projectile Motion (or You Bet Your Grade)

Roy Coleman                      Morgan Park High School
                                 1744 W. Pryor Ave.
                                 Chicago, Illinois 60643
                                 312-881-5050

Objectives:

To calculate the initial velocity of a spring stretched a varying amount and 
fired vertically from a shooter and to predict, by calculations, where that 
spring will land when it is shot horizontally off of a horizontal platform.  As 
a additional objective, to learn that ALL data must be recorded but that data 
which is way out of the ordinary can be ignored when calculating averages and 
results. 

Materials Needed:

The following materials are needed for each group of 2-4 students:
  1) a spring with sufficient elasticity so that stretching it about 1 cm will 
     shoot it to a height of about 50 cm.  One end should have a paper clip 
     attached and the other should be bent into a loop which will hook over the 
     nail in the end of the shooter described below  (Primary level students 
     might use rubberbands instead but they are not as consistent.)
  2) a 'shooter' made from a 1" by 1" by 18" piece of wood with a headless nail 
     driven into the end near the edge so that the spring may be hooked to it
  3) several metersticks
  4) graph paper
  5) a section of a room with sufficient height where a spring's shot may be 
     marked and/or measured directly
  6) optional - an apparatus which simultaneously drops one object while 
     projecting a similar object horizontally

One of each of the following is needed for the entire class:  
  1) a securely mounted horizontal surface at a height of 1.5-2 meters above the 
     floor where the students will hold their 'shooters' for the You Bet Your
     Grade part of the experiment
  2) a low waste paper basket

Strategy:

INSTRUCTORS PREPARATION:
 1) Mark the 'relaxed' position of the spring on the side of the shooter.
 2) Stretch and shoot the spring vertically such that it goes to a height of 
about 50 cm.  Make enough shots so that you are confident of the measurements.  
Mark this stretch on the side of the shooter and number it #1. 
 3) Stretch and shoot the spring so that it goes to the greatest height that you 
are going to use.  Again, make enough shots so that you are confident of the
measurements.  Mark this stretch on the side of the shooter and number it #6.
 4) Measure the distance between the two marks (#1 and #6) and divide it by 5.
Measure down from #1 a distance equal to the result and mark that point #2.  
Repeat the measurement for points #3 and #5, intentionally skipping where #4 
should be.  (A suggestion was made to drill holes at each of these points to 
allow a small nail to be inserted and the paper clip hooked on the nail.  
Pulling the nail fires the spring.)

CLASS ACTIVITIES:
 1) Have the students fire the spring vertically from each of the indicated 
stretches.  They should record ALL of their data and decide which of the points 
are consistent and which are in error.  (Does the height get more consistent as 
the student gets more experience?) (Does it make a difference if a different 
person does the shooting?)
 2) Plot a graph of height vs. stretch.  (Is this graph a straight line?) (Does 
it go through zero?)                                     
 3) Calculate the initial velocity of the spring (vo=sqrt(2gh)) for each of the 
stretches and plot a graph of the initial velocity vs. stretch.  (Is this graph 
a straight line?)
 4) Demonstrate that a projectile fired horizontally will take the same amount of 
time to reach the ground regardless of its initial horizontal velocity.  This 
should demonstrate that the vertical acceleration of an object is independent of 
the horizontal acceleration with the resulting motion being the sum of the two. 
 5) Show the students the board you have mounted horizontally and how their 
shooter should be placed.  Also show them the waste paper basket that they will 
be shooting into.
 6) Have the students predict, from their graphs, the initial velocity of their 
spring for a stretch equal to what would be #4.
 7) Have the students calculate the time that it will take for the spring to 
fall from the board to the top of the waste paper basket.  (t=sqrt((2H)/g))  For a 
high school class, you might just tell them to calculate the time to fall and 
let them figure out that the spring doesn't fall to the floor but only to the 
top of the basket.
 8) The students should then calculate where to place the basket.  (R=vot)
 9) Place the center of the basket at the calculated place and have them try 
shooting.  (Did they hit it and how many tries did it take?)

Expected Results:

    The results for the first few shots should be somewhat inconsistent but, 
after 5-10 shots, they should improve greatly.  Generally the shots should group 
within about 10% of the average value for each stretch.

    Changing the person doing the shooting usually makes a noticeable difference 
because each person has their own unique way of releasing the spring.  Using a 
shooter with holes and a nail release along the side should eliminate this 
difference. 

    The graph of height vs. stretch should, with the possible exception of the 
first point, be a straight line.  It will probably not go through zero since a 
spring is linear only from the point where all of the coils are separated and it 
usually takes some small initial stretch to completely separate all of the 
coils.

    The graph of velocity vs. stretch should NOT be a straight line because of 
the square root used to calculate it but, for the small sample of points tested, 
it will probably be very close to a straight line.

Evaluation:

    A student's grade is based on how many shots it takes for the spring to land 
in the basket.  (First shot = A, Second shot = B, etc.)