Recall that you used a tilted airtrack and a sparker to study one-dimensional motion with constant acceleration. In this lab, you will use a tilted air table and a sparker to study two-dimensional projectile motion. We will refer to the two dimensions involved as "horizontal" and "vertical" even though they aren't literally so. You will perform two experiments on your tilted air table:

1) a "drop," in which a puck released with no initial speed on the tilted air table will accelerate in one dimension down the table

2) a "launch," in which the same puck will be propelled (by your hand) as a projectile. In this case, you will give the puck both an initial upward vertical velocity (vi,v) and an initial horizontal velocity (vi,h) on the same tilted air table

GOAL:

One goal of the lab is to determine if the horizontal motion of the launched puck has any effect on its vertical acceleration. Another goal is to compare the measured values of the range, height, and time length of the launch with values predicted by the d-v-a-t formulas.

READING: Giancoli Chapter 3, especially example 3-8

PRELAB:

In the case of the two-dimensional launch, where acceleration acts in one of the dimensions (the vertical, v), but not in the other (the horizontal, h), the equations for the displacement in the h and v directions are:

dh = vi,h t and dV = vi,v t + 1/2 avt2

The first good spark has coordinates: (dh=0, dv=0). Set t=0 at this spark.

1) Do the necessary algebra to eliminate the variable t from the system of equations above and solve for dv as a function of dh and the "known" constants (vi,h, vi,v, and av). You will thus be finding the shape of the launch trajectory. [Think of the initial speeds (vi,h and vi,v) and the acceleration (av) as pure numbers which will be found in the same manner as for the position-vs-time graph in the air-track lab (using the TI-82 to fit a quadratic).]

2) What do mathematicians call the shape of the curve represented by this equation?

3) Use the appropriate d-v-a-t formula to solve for each of the following quantities in terms of the known quantities (which are vi,h, vi,v, and av). Again, assume that the initial position ( t = 0 ) of the puck is at dh = 0, dv = 0 :

a) the time it takes the puck to reach maximum height. Hint: what velocity condition occurs at maximum height?

b) the maximum value of dv (i.e., the maximum vertical height that the puck reaches) during the launch trajectory

c) the horizontal distance (also called the "range") that the puck travels from launch until it again returns to dv = 0

PROCEDURE:

Operation of the air table will be demonstrated, and you will have an opportunity to practice using it before taking data. The air table will be tilted with respect to the horizontal lab table by some small angle, so that a puck placed near the upper edge will drift toward the lower edge with a constant acceleration.

IMPORTANT NOTES:

1) Do NOT touch the table, the air hoses, or the pucks while the spark generator is running. You will get a substantial shock (more surprising than painful) if you do. It is not dangerous unless you have problems with your heart rhythm.

2) Both pucks must be on the table and connected to the spark generator for sparks to be produced. During the launch and the drop, leave the other unused puck in of the lower corners of the table.

3) Place the large sheet of carbon paper directly on the table and then place the large white sheet of recording paper over that. We suggest that the latter be placed so that it is flush against the lower edge and one side of the table.

4) The foot pedal (or hand switch) must be held down continuously for the spark generator to operate. When you release the pedal, the sparks stop.

5) Obtain spark records for the launch and the drop on the same side of the paper. Since the launch is harder to execute than the drop, perform the launch first; once you obtain a successful launch spark record, then do the drop. Do NOT move (or completely remove) the paper in between the launch and drop! You need only gently lift and turn the top part of the paper back to make sure that you have sparks, which will show up only on the side of the paper facing the carbon sheet.

6) Make sure to record the spark rate (we suggest 10 Hz) and the height of the riser block that is used to prop up one leg of the air table.

THE LAUNCH

For this run, start the puck near a bottom corner and launch it upwards and also toward the bottom corner on the opposite side of the table. The goal is to have the puck launch path cover as much of the paper as possible. The puck should hit the bottom or the side of the table near the bottom opposite corner

Stop the spark generator just before the puck hits the bottom or side.

THE DROP

Place the puck near the top center of the table (but completely over the carbon paper). Release the puck and then quickly start the spark generator. In releasing the puck, refrain from giving it any sideways motion. Again stop the spark generator the instant before the puck hits the bottom.

As soon as you remove your paper from the table, label it with your names, type of run, apparatus (letter) used, initial and final positions of the launch.

RESULTS

1) CHOOSING AXES

As in the air track lab, you may ignore one or two of the initial sparks if you feel that it is appropriate; on the other hand, you may not need to.

a) Define the initial sparks for both the drop and the launch to be the origin

(dh = dv = 0 ).

b) Choose your vertical-axis to be parallel to the drop trail of sparks; see diagram on last page.

c) Draw a second vertical-axis (parallel to the first one!) through your launch origin. Now draw an horizontal-axis through your launch origin that is exactly perpendicular to your vertical-axis. Extend the horizontal-axis far enough so that it intersects the final part of the launched puck's path. (If your trajectory does not extend down to your horizontal-axis, you may need to choose a different spark as your origin. Consult the instructor in this case.)

d) Choose and label a positive direction for both horizontal and vertical. Stick with that choice of direction for BOTH the drop and the launch. You should, however, choose different origins for the drop and the launch.

2) DIRECT MEASUREMENTS OF LAUNCH CHARACTERISTICS

Carefully measure (i.e., use a ruler) and record the following three quantities directly from the launch trajectory. Pay careful attention to significant figures. Even though the sparks were produced at tenth-of-second intervals, you can measure time to the nearest hundredth of a second by interpolating. Record any positions and distances to the nearest half a millimeter.

a) maximum vertical height reached during the launch.

b) time required to reach maximum height

c) the range (horizontal distance) covered by the launched puck

3) PREDICTED LAUNCH CHARACTERISTICS

In this part you will determine values for these same three quantities from the d-v-a-t formulas (derived in the Prelab) and the initial and ambient conditions (vi,h, vi,v, and av) for the launch.

Measure the values of horizontal and vertical displacements for each launch spark, and record them along with the time-value t in an appropriately labeled, organized data table. Be consistent with the directions that you defined as positive earlier.

Use the TI-82 calculator to perform an appropriate fit to your launch data in order to determine the values of vi,h, vi,v, and av. No calculations of instantaneous velocity (as in the air track lab) are necessary.

Once you have obtained values of vi,h, vi,v, and av, use them along with the formulas derived in Prelab to calculate values for the maximum height reached, the time to reach maximum height, and the range.

4) ACCELERATION OF THE DROP

Measure and record the vertical displacement values and times of each drop spark in an appropriately labeled, organized data table.. Remember the previously-made choice of origin and positive direction.

Again use the TI-82 to perform an appropriate fit to your data in order to determine values of vi,v, and av for the drop.

Your report should include 3 computer-printed graphs; each graph should also have an associated set of fit parameters (a, b, c) also printed by computer. In order to identify the physics quantities, write down the math equations of the fit, the physics equations that they represent, and then an appropriate translation table (or matchup) as you did the air track lab.

Also include your air table spark record sheet. It should be very neatly folded (with the partner names on the outside).

ANALYSIS

1) Compare the vertical accelerations for the drop and the launch. ["Compare" from now on in Physics Lab means "write the two values down side-by-side and find the % difference between them."] Should the two different values of ay be identical? Why or why not?

2) In a table, list the directly measured values of the maximum height reached, time to maximum height, and the horizontal range along with those calculated from the d-v-a-t formulas and the graphically determined values of vi,h, vi,v, and av. Include the percent difference in each case.

SUMMARY