Lab 7: Trajectories 3/23/15

Physics Lab #7 started on 3/23/15
Trajectories
Annemarie Branks
Professor Wolf

Objective: Predict where a ball, flying off of a track, will hit a board angled below the track.

Procedure:
     1.  Securely set up the apparatus using a liberal amount of masking tape to keep the track static. Mark a part of the track where you will launch the ball from for each trial.

     2.  Using a weight tied to a string, mark the floor with tape to indicate where the ball leaves the track. You will use this marking to measure how high the ball is from the ground when it launches and how far it launches in the x-direction.
     3.  Tape carbon paper to the floor. Release the ball from the same place five times and measure how far in the x-direction the ball launched. Our average horizontal distance is 50.5 cm +/- 0.3cm. 

     4.  Knowing the distance in the x-direction, and having measured the distance in the y-direction (94.2 +/- 0.1 cm) we can determine how fast the ball is leaving the track. This is how we solved for V0x:

     5.  Set up a wooden board directly below the track at angle. We are going to solve for the distance the ball traveled in relation to the board. We need to know the board's angle, the height from which the ball launches, and the ball's initial velocity which we found from step 4. Here are the calculations for our experiment:

     6.  Cover the board with carbon paper. Then launch the ball 5 times and measure the distance from the track. Make sure neither the board nor the track moves at all.

     7.  Measure how far on the board your ball made its marks and see if your actual data matches up with your calculated data from step 5. Our experiment gave us an average distance of 48.02 +/- 0.52 cm. Your numbers may not match up due to uncertainties in measurements. Calculate the range on uncertainty for d. 
Our equation is d = (2*V0x^2sinα)/(g(cosα)^2) which, if you trace back your steps in calculating d, you will find simplifies to (sinα*x^2)/(ycos^2α). This can also be written as (x^2/y)tanαsecα. We are going to use the second equation so we can use product rule when it comes time to derive. 

d = (x^2/y)tanαsecα

dd = |∂d/∂y|dy + |∂d/∂x|dx + |∂d/∂α|dα

dd = |(-x^2/y^2)tanαsecα(0.1)| + |(2x/y)tanαsecα(0.3)| + |(x^2/y)(secαtan^2α+sec^3α)(0.035)|

dd = 0.133+0.915+5.267 = 6.315

With our uncertainties in measurements, the distance the ball traveled in reference to the board should be 47.47 +/- 6.315 cm. This range is fairly large which is not ideal but it matches up well with the actual average distance of 48.02 +/- 0.52 cm.