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LAB 5:  Projectile in Curvilinear Motion

Purpose:  (Write a statement, based on our discussion in class)

Materials:  (list them)

 PART I:  Horizontal Projectile From a Height.

Diagram:  (Using the diagram shown below as a model, construct a diagram showing the paths taken by the cannon ball for both forces applied.  Include all required labels, including greater detail for the components of S_H)

                  (Diagram 1)

Click on button for more information:    

Data Table:

* time values should be rounded to the nearest 0.1 second.  Rounding helps to compensate for our inability to react within a span of 0.01seconds, and allows for a more meaningful comparison of t_EXP and t_MATH values.

Procedure:   

(1)   The cannon should be attached to its support base at 0° inclination (Photo 1).  Practice arming the UNLOADED cannon by pulling back the white knob at the end opposite the plunger, until the tension adjustment  nut on the plunger slides past its counterpart on the stationary shaft (Photo 2).  The flat faces on each nut should now be in contact. DO NOT PLACE YOUR HANDS OR FACE IN FRONT OF THE PLUNGER.   Pulling upward on the silver ring attached to the stationary shaft will cause the two cylindrical nuts to separate, resulting in the plunger moving forward WITH GREAT FORCE (Photo 3).  The cylindrical tension nuts can be adjusted to produce a greater or lesser force behind the cannon ball.

(2)   Check that the cannon is firmly attached to its support stand, by twisting the black retention knob clockwise.   The apparatus must be brought to the edge of the lab table, facing the center of the room.  The apparatus must not shift left or right from one shot to the next, so use a C-clamp to hold the support base to the table top.  The cannon must remain oriented in a way to permit the cannon ball an unobstructed path to the floor, so classroom chairs must be removed from the vicinity of the ball's flight path. 

(3)   While the cannon is in the uncocked position, hold the free end of the plumb bob's string against the cannon ball.   The plumb line should be centered on the front of the cannon ball, as the bob, suspended a millimeter from the floor, is brought to rest.   Place a piece of masking tape on the floor directly beneath the plumb bob, and sketch a line on the tape to denote this exact point.  This position will represent the "origin" of the cannon ball, from which S_H(EXP) is measured (Photo 4).   Measure the vertical distance from the mark on the floor to the center of the cannon ball.  This distance represents S_V (Careful!  This dimension exceeds one meter, so two metric sticks or a carpenter's rule will be required). 

(4)   The first run requires minimum force behind the cannon ball (check with the teacher), so move the cylindrical tension adjustment nut on the stationary shaft forward, then lock it into place by turning the smaller hex nut until it firmly makes contact with the tension nut.  Place the cannon ball on the spindle of the plunger.  TAKE NOTE OF THE IDENTIFYING NUMBER WRITTEN ON THE CANNON BALL.      

(5)   Allow the ball to propel forward, and take note of its landing site.  Tape a blank sheet of paper at this location.  The edge of the paper nearest the cannon ball must be marked with an arrow, to help distinguish the edge from which S_P-D  (i.e., paper edge-to-dot) distances are read (Photo 5). Also measure the distance from the mark on the masking tape representing the "origin" of the cannon ball, to the edge of the blank sheet of paper.  This distance is recorded as S_C-P (i.e., cannon ball-to-paper).

(6)  Once a sheet of carbon paper is laid over the blank paper, the ball should be shot at least five times, until a discernable pattern is recognized.  The precise landing points of the ball can be located and measured (Photo 6).  The average of these marks is recorded as S_{P-D (AVG)}.  Each dot must be labelled with its S_P-D value.  Include this paper with your lab.

(7)  The time of flight (t_EXP) can be taken with a stopwatch, by a lab partner having the best hand-eye coordination.  This person  should also release the ball, by pulling on the silver ring at the moment the stopwatch start button is pressed.

(8)  Repeat the procedure, propelling the ball with the greatest force allowed by the cannon.  This tension must be used during Part II and Part III, so secure the tension adjustment nut firmly.

Sample Calculations:   You must show one sample representing each "MATH" derived value (t,V_V, V_H, V_S) for one of the two sets. 

        Show ALL steps in each solution, including appropriate subscripts.

        Use the following guidelines when calculating your mathematically derived values:

        t_MATH  -  use S_V

        V_H(MATH)  -  use S_H(EXP) and t_MATH

        V_V(MATH)  -  use S_V

        V_S(MATH)  -  use V_H(MATH) and V_V(MATH)

PART II:    Projectile on a Parabolic Path From Point to Point on the Same Horizontal Plane.

Diagram:   

                      (Diagram 2)

Data Table:

  * Write the exact angle of inclination as measured by you in this cell.

Procedure:    

     (1)   Remove the cannon from its support stand by twisting the black knob of the retention bolt counterclockwise.  Once removed, you will note a series of holes in the side of the support stand.  Place a piece of paper behind the holes and outline the holes as circles on the paper by placing the lead tip of a sharpened pencil through each hole.   Using the solo circle as the origin (from the threaded hole into which the retention bolt is screwed), use a straight edge to draw long rays through the center of each of the other circles.   The angle between the horizontal and each of these two rays must be precisely measured using a protractor, then recorded in the data table (Photo 7).

    (2)   Firmly reattach the cannon so that it is inclined closest to 45° ; the cannon must not wobble about its attachment.   The projectile apparatus must be placed on the floor.  While the cannon is in the uncocked position, hold the free end of the plumb bob's string against the cannon ball.   The plumb line should be centered on the front of the cannon ball, as the bob, suspended a millimeter from the floor, is brought to rest.   Place a piece of masking tape on the floor directly beneath the plumb bob, and sketch a line on the tape to denote this exact point (Photo 8).  Measure the vertical distance from the mark on the floor to the center of the cannon ball.  This distance represents S_V. 

    (3)   The mark on the masking tape represents the "origin" of the cannon ball,  from which S_H(EXP) is measured.  Before S_H(EXP) can be measured, the cannon ball must be test-shot, so its landing site can be identified.   The landing site must be at the same elevation as the uncocked ball (Photo 9).  This is accomplished by standing a cardboard box on end, then placing a ceramic pad on the box. The box can be stabilized by placing several textbooks in the box.  A blank piece of paper is then taped to the ceramic pad, with a sheet of carbon paper on top, so impressions of the impacting ball can be recorded.  Further adjustments can be made to the height of the landing site by placing a thin book or two beneath the ceramic pad.   The edge of the blank paper nearest the cannon should meet the very edge of the ceramic pad, which should be fixed at the very edge of the box (Photo 10). 

    (4)   Make sure the cannon ball is shot while the cannon's spring tension is at the maximum setting used in Part I of the lab.   Measure the distance between the mark on the tape at the "cannon ball's origin" and the nearest edge of the white paper (also the nearest vertical surface of the box). Record this dimension as S_C-P.   The ball should be shot at least five times, to generate an average for time of flight, t_(EXP), and an average for the distance from nearest edge of paper to dot, S_P-D.  S_H(EXP) is equal to the sum of S_C-P and S_P-D(AVG).

Sample Calculations:   You must show the math steps leading to each "MATH" derived value (R, H, and t).              

        Show ALL steps in each solution, including appropriate subscripts.

        Use the following guidelines when calculating your mathematically derived values:  

        V_{O (MATH)} -  use R_EXP and theta

        H_MATH  -  use standard 'H" equation for parabolic path, including V_O(MATH)

       R_MATH  or S_H  -  use V_OH  and  t_MATH

        t_MATH  -  use V_O(MATH) and standard "t" equation for parabolic path to same horizontal plane

PART III:  Inclined Projectile from a Height

Diagram:    (Using Figure 3 below as a model, construct a diagram showing the approximate path followed by the projectile during each of the two runs, including all the required labels and details).

                     (Diagram 3)

Data Table:

* Have you adjusted S_V at 45° by placing a thick pad beneath the sheet of paper at the landing point?  The height of this pad must be subtracted from the overall height of the ball above the floor - this adjusted value is "S_V" at 45° and is very close to the S_V measured at 23°.

  Procedure:

     One goal of Part III is to shoot the cannon ball at two angles, while retaining a constant vertical height, S_V, between ball and floor.  In order to accomplish this, the distance from ball to floor is measured at both angles.  This preliminary procedure requires you to attached the cannon at the higher angle, measure the ball's height, then reset the cannon at the lower angle, and measure the vertical height again.  The difference between the two values for S_V is the thickness of the "padding" that must be placed beneath the blank sheet of paper positioned on the floor at the ball's landing site AND the masking tape marking  the ball's origin when the cannon is set at the higher angle.* 

    (1)   The cannon should be inclined at 45°, as set in Part II.  Check that it is firmly attached to the stand so that it does not wobble about its attachment.  Retighten the clamp holding the support base to the edge of the lab table.  The cannon must remain oriented in a way to permit the cannon ball an unobstructed path to the floor (classroom chairs were removed during Part I).  The apparatus must not shift left or right from one shot to the next.

    (2)   While the cannon is in the uncocked position, hold the free end of the plumb bob's string against the cannon ball.   The plumb line should be centered on the front of the cannon ball, as the bob, suspended a millimeter from the floor, is brought to rest.  Place a piece of masking tape on the floor directly beneath the plumb bob, and sketch a line on the tape to denote this exact point. Using two metric sticks placed end-to-end, measure the vertical distance from the mark on the floor to the center of the cannon ball.  This distance represents S_V. 

    (3)   The mark on the masking tape represents the "origin" of the cannon ball,  from which S_H(EXP) is measured.  Before S_H(EXP) can be measured, the cannon ball must be test-shot, so its landing site can be identified.  Make sure the cannon ball is shot while the cannon's spring tension is at the maximum setting used in earlier parts of the lab.  Once this has been accomplished, tape a sheet of blank paper covered with carbon paper at the landing site.   Measure the distance between the mark on the tape at the "cannon ball's origin" and the nearest edge of the white paper.  Record this dimension as S_C-P.   The ball should be shot at least five times, to generate an average for time of flight, t_(EXP), and an average for the distance from nearest edge of paper to dot, S_P-D.  S_H(EXP) is equal to the sum of S_C-P and S_P-D(AVG).

    (4)   Repeat the procedure for collecting data, once the cannon is reset at the angle closest to 30°.   If you intend to use the tape on the floor marking the cannon's origin as this run's origin, you will have to unclamp the apparatus, move it backward, until the plumb bob indicates that the ball is directly over the mark on the tape (as the cannon's inclination is reduced, the ball is shifted slightly forward).

Sample Calculations:  You must show the math steps leading to each "MATH" derived value ( t, V_O, V_H, V_V, and V_S) for both angles.  Show the work for each variable side-by-side. 

         Use the following guidelines when calculating your mathematically derived values:

        t_MATH  -  use S_H(EXP)

        V_O(MATH)  -  use S_H(EXP), t_EXP, and theta

        V_H(MATH)  -  use V_O(MATH) and theta

        V_V(MATH)  -  use V_OV(MATH) and S_V(EXP)     (caution: is S_V a positive or a negative value?) 

                 [Ha, Ha!  How are you going to get V_OV(MATH)?   hint: use only (EXP) variables for height and time]

        V_S(MATH)  -  use Pythagorean theorem

Conclusion:

(A)  Part I (horizontal projectile from a height):

    For each term listed, discuss the observed result.   Explain what should have happened and why.   Provide actual data and include the formula used to determine the result:

        As V_H is increased,

            (1)  t_TOT ?

            (2)  V_FV ?

            (3)  S_H ?

            (4)  V_S   ?

            (5)  V_OV ?

(B)  Part I (horizontal projectile), Part II (parabolic path), Part III (inclined projectile from a height):

     Assuming the projectile in each of the three sections of the lab had a similar maximum force from the cannon (force_max), explain the relationship between the terms in the following pairs (i.e., lesser than, equal to, greater than).  Using appropriate formulas and properly subscripted variables, explain precisely why differences exist between the two terms, or why both terms are equivalent.  Include actual data:

            (1)  t_T(fig.1)   vs.  t_T(fig.3)

            (2)  V_S (fig.1) vs. V_S (fig.2)

            (3)  V_S (fig.1)  vs. V_S (fig.3)

            (4)  S_H (fig.1)  vs.  S_H (fig.3)

            (5)  S_H (fig.2) vs. S_H (fig.3)

            (6)  V_S (fig.2) vs. V_S (fig.3)  (IGNORE THIS PROBLEM)

(C)  In Part II, if V_{O (MATH)} is calculated using t_EXP, why can't t_MATH be calculated using V_{O (MATH)}?  Show both formulas in your explanation.

(D)  Assuming the V_O  is the same maximum value for both runs, use the data from Part III to calculate the horizontal range (R_MATH) for each projectile at the moment the projectile reaches an altitude of S_V again on its downward journey.  State both range values, and explain the difference in magnitude between the two.  Show all work.

    State the two S_H values once the projectile strikes the floor.  Explain the difference in magnitude between the two values.  Also, compare the two paths of the projectile.  Is there any point where the two paths intersect?  Explain the significance.

(E)  Using the experimental data from Part III and formulas learned in class, can t_MATH be derived independent of t_EXP, to avoid  t_MATH = t_EXP caused by interrelated terms?   Whether it is possible or not, your explanation must be substantiated mathematically through the manipulation of various terms.  (IGNORE THIS PROBLEM))