Lab 5 - March 27, 2015 - Trajectories

Lab 5
Brandon Elder
3/27/2015

Trajectories

Purpose: To determine the impact location of a ball launched from a ramp onto a slanted platform that will be placed at the base of the metal ramp. The location will be determined by observing the ball's distance first without the slanted platform and then using kinematic equations to determine the impact location with the platform. 

Fig. 1 The apparatus was set up
as seen above. The top
metal ramp was taped
to the bottom ramp. The angle
does not matter for the two
metal pieces so long as it
remains the same.

Fig. 2 Carbon paper on the
ground with the paper to
mark the location of the
ball drops.

Procedure: Set up of the apparatus should be similar to Fig. 1. Place a piece of paper on the ground and use tape to keep it in place. The paper should be in the location that you expect the ball to land at. Let the ball drop from the apparatus once to estimate the correct location for the paper.  Next, tape a piece of carbon paper to the paper so that the location of the ball falling on the paper will be marked from the carbon (see Fig. 2). Once the correct location has been determined and the paper has been taped, choose a starting location for the ball on the top metal ramp to drop the ball from. It is important that the starting location for the ball is the same each time data is collected. This will allow for relatively the same location being marked on the paper as the ball will launch off the ramp with roughly the same velocity. Place the ball and let it roll down and launch it off five different times. The ball will hit the carbon paper and mark a dot on the paper. After five runs there will be five dots on the paper all in relatively close proximity (see Fig. 3). Watch the video below to observe exactly how the apparatus works and notice the ball bounce right on the paper with the carbon taped to it.

Fig. 3 Five dots on the paper relatively
close with one dot a little off.
 This one dot is off because we
ran a 6th trial with the ball at a different
starting location. This proved to
 us that it was critical to have the
starting location be the same for all the runs. 

Data Analysis:

Fig. 4 Diagram of apparatus with the
measured height and horizontal distances.

The next step in the process will be to analyze the test results. Measure the horizontal distance on the ground from the base of the ramp edge to the center of the five dots. Record this measurement. Measure the vertical distance from the edge of the ramp to the floor. Record this measurement. Then draw a diagram with the measured distances inputted (see Fig. 4). 

Next, using kinematic equations, calculate the time that it took the ball to travel and hit the ground. Once you have the time, you can calculate the speed at which the ball is launching from the ramp. The equation used and the calculations completed can be seen in Fig. 5.

Fig. 5 The equation used to determine
 the time and then the equation used
 to determine the velocity.

Fig. 6 Set up of board to the
apparatus. It is important to note
that the board must rest right up
against the edge of the ramp
from the first part of the experiment.

Now that you have calculated the time and the velocity at which the ball left the ramp, you can perform the next part of the lab. The next part of the lab will be to attach a slanted board to the end of the apparatus and to determine where along the board the ball will land. Measure the angle at which the board is laying against the apparatus. See Fig. 6 for the set up of the new board to the apparatus. The object here is to derive an expression that would allow you to determine the value of "d" (see Fig. 7) given that you know the initial velocity and the angle at which the board is slanted. The formula will look like Fig. 8. This gives you an expression for "d" in terms of the velocity.

Fig. 7 "d" is the distance down the ramp that we are
estimating with our formulas from below.



Fig. 8 Calculations used to determine the formula for "d"     

However, velocity is an expression that came from the x distance measurement as well as the y distance which was used to calculate time. The formula must be re-written in terms of all calculated expressions. The calculated measurements were the x-distance, y-distance, and the angle of the ramp. Re-writing velocity results in the expression below, (see Fig. 9).

Fig. 9 The formula for "d" in terms of theta, x, and y.

All that is left at this point is to plug in numbers to decide what the distance of "d" will be, and THEN calculate the uncertainty of course. We just need to take partial derivatives of the equation that we have for d with respect to the three measurements (theta, x, and y). Each partial derivative result will be multiplied by the known uncertainty in each measurement. The sum of all three products will result in the total uncertainty in our calculations. Finally, perform the experiment once more and measure the distance down the board to determine the actual distance. Compare that value to the value that you calculated. Is the data within the amount of "+/-" uncertainty? If so then you have a pretty good proof of just how accurate this experiment was.

Conclusion:

Our data gave us a calculated result of 1.12 meters for how far the point of impact down the board "d" should be. Our uncertainty calculated out to be +/- .0068 meters. When we actually ran the experiment and measured the data we recorded the ball landing at 1.06 meters down the board. This is not within the amount that we had calculated. :(

After recovering from a few days of severe depression following the realization that our experiment had failed, I started to reflect on possible issues for this non-concurrence. One main issue was that the board had been bumped and re-adjusted in-between the measurement of the angle and the performing of the experiment. This would have totally caused the type of error that we witnessed in our final results. The calculations for "d" were determined after we measured our angle theta but before we ran the actual experiment.