Analysis

Theoretically, our pellet should decelerate in each of the fluids according to the equation

If we plot position against time for the particle in the fluid, we use a mathematical analysis program to fit a logarithmic curve of the form

The two constants that we get out of the fit equation A and B correspond to 1/K and K*v_i.  If we know v_i then we can find K.

If we find K for the pellet in water, and find K for the pellet in water after passing through oil, we may be able to associate any significant differences to the interface between the two liquids.

Picture Analysis

In each picture, there is a known distance which we use to scale our measurements on the picture to the distances they correspond to in actuality.  In Picture A, the scale factor is the width of the glass plate at the bottom of the tank.  When setting the scale, it is important to measure this distance in the same plane in which the pellet travels.  In Picture B, the scale is the distance between the oil level and the water level.  In Picture A, initial velocity is assumed to be the same as in picture B.  In each picture, we measure the distances from the surface of the water to the position of the pellet at each flash.  We then apply the scale factor to these values to get the actual distances the pellet has traveled.  We know the time which corresponds to each flash, and thus can plot position vs. time of the pellet and then apply our fit.

Picture A

Picture A was not useful for the comparison part of our analysis, but it does provide a measurement of the initial velocity which we can use to analyze Picture B.

 

First we measure the distance between the oil and the water and call it

scale_pic

scale_pic=6.70cm

We know that this corresponds to a real distance distance which we call scale_real

scale_real=15.5 cm

Dividing scale_real by scale_pic and we get a scale factor of 2.31

In all of our pictures we measure on the picture the distance traveled by the pellet which we label Distance_pic

Distance_pic= 5.15cm for the pellet in the air in Picture A

Applying the scale factor gives us Distance_real.  Then dividing by the time interval of .499ms yields a speed of 239 m/s

Picture B

Picture B shows a pellet which is traveling through water only and will be used for comparison.  In this picture Distance_pic is measured from the “origin” which is the surface of the water.

Using the glass plate as our scale, we get a scale factor of 1.22.

   Time    (s)                         Distance_pic      (m)               Distance_real   (m)

0	0.019	0.023
0.000288	0.060	0.073
0.000576	0.089	0.108
0.000864	0.114	0.139
0.001152	0.132	0.161
0.001440	0.147	0.180
0.001728	0.162	0.197
0.002016	0.175	0.213
0.002304	0.187	0.228

  

 

 

 

 

 

This graph shows a plot of position versus time with the fit

A = 1/K and AB = v_i

A = 0.121 m and K = 1/.121 m = 8.26 m^-1

B = 1900 s^-1 and v_i = (1900 s^-1)(0.121 m) = 230 m/s

These fits produce values of 8.26 m^-1 and 230 m/s for the constant K and initial velocity and respectively.

 Picture C

Picture C is much like picture A except it has a distribution of images that allows us to analyze the pellet in the oil and in the water.  There are two different fits on the graph, one for the first three data points which are in the oil, and the last three which are in the water.

 

Time (s)	Distancepic      (m)	Distancereal   (m)
0	0	0
.000499	.0380	.0843
.000998	.0651	.144
.001497	.0810	.180
.001996	.0953	.211
.002495	.106	.236

Oil                           Water

A=.135 m                 A=.192 m

B=18100 s^-1             B=686 s^-1

1/A = 7.4 m^-1           1/A=5.2 m^-1 

AB= 244 m/s           AB= 131 m/s

These fits yield an initial velocity in the air of 244m/s, a K of 7.4 m^-1 for the oil, a velocity of 131m/s across the oil/water interface, and a K of 5.2 m^-1 for the water after passing through the oil.