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Instantaneous Velocity

Instantaneous velocity, in one dimension, is defined as the velocity at a particular instant of time. It is mathematically stated as:
 

v(i) = lim/t --> 0 x/t

(1)

where the small distance, x, through which an object moves in some small time, t, is measured and the ratio is computed as the time interval becomes infinitesimally small. This is basically the derivative or the time rate of change of position. This is not a simple task. In practice, since timers frequently use vibrations as a comparative in making time measurements, as t becomes smaller than the vibration, or at least of roughly the same size, we see that making a reasonable comparison with that device is unreasonable. You will find a similar problem in this experiment. In fact, it really is impossible to measure things to the limit as the independent variable goes to zero. In practice then, we will determine this quantity in a somewhat indirect manner.

You will use a Photocell Gate to measure the time interval. Be sure to read Appendix , entitled "Collecting Data With a Photocell Gate." This information is reproduced directly from the manufacturer's notes and is most complete. This should prepare you to use the device in measuring time.

The Photocell timer works so that a beam of light from the source to the phototransistor allows an electric current to pass through the circuit. When the light is interrupted, no current passes. Thus as an object passes in front of the detector, time is measured.

Figure 1: Object of length X passing in front of timing gate.

Consider an object of length X passing in front of the detector. See Figure 1. Assuming that the object is weightless compared to the slider to which it is attached, it has a certain initial velocity as it first arrives at the point where the time measurement is made. If it accelerates at a constant rate in front of that point, then the width of the object divided by the time the object interrupts the circuit as indicated on the timer, gives us the average velocity of the object as it passes the timer. We are interested in, however, the instantaneous velocity of the object as it first crosses the timer. This is the limiting case where the time becomes infinitesimally small. So does the width of the object.

We cannot, however, use a flag so small that it approaches the size of the timing gate. The recorded times become extremely unreliable.

If we use smaller increments of distance, we can get better approximations of the instantaneous velocity. Be careful to note that we are still measuring an average velocity over a small interval. The instantaneous velocity is the limit of the average velocity as the time becomes very small, approaching zero. But, if we plot these average velocities vs. t, we can see what happens when t becomes zero by extrapolation. One requirement as we make these measurements repeatedly is to be able to consistently reproduce the same velocity.

Figure 2. Linear Regression performed on Excel.

One method to consistently reproduce a velocity for measurement repeatedly is to use uniform acceleration. The acceleration due to gravity is an obvious choice. This also allows us to make an estimate of the acceleration due to gravity. In the measurement of instantaneous velocity it is not necessary to even know what the acceleration is, so long as it is consistent.

The relationship between the instantaneous velocity, the initial velocity, the acceleration, and the time is:
 

v = vo + at

(2)

where v = instantaneous velocity, v_o = initial velocity, a = acceleration, and t = time elapsed. As long as the acceleration is uniform (constant) the result of the curve of velocity vs. time should be a straight line (y = mx + b or v = at + v_o ). The acceleration can be computed from the slope of this line.

With the air track inclined, the acceleration that the object receives is a component of the acceleration due to gravity. This is how Galileo first measured the acceleration due to gravity. Physicists today might use a pendulum and relate the acceleration, length, and period, but we can get some very good results. If you compare this value with the accepted value in a Physical Handbook (CRC or others) you can compute your percent error.

Figure 3: The acceleration a is a component of g.

OBJECTIVES:

1. To measure the instantaneous velocity of an object and the acceleration due to gravity. Motion is studied on a frictionless inclined plane.
2. To practice indirect methods of determining a quantity.
3. To develop a linear regression curve for an appropriate set of data using EXCEL.

APPARATUS:

• 1 Photogate Timing System
• 1 Air Track System
• Miscellaneous Clips and Flags
• Ruler and Magnifying Glass
• LINREG computer program

PROCEDURE:

1. Level the air track and then tilt it around 10 and 15 mrad. Record this angle for future use in calculating the acceleration due to gravity. Estimate the uncertainty in the determination of this angle. Record the angle q.
2. Set up the photocell gate and timer. See Note 1. Try it a few times until you feel comfortable that it works properly under your control. As you "play" with this system you should notice that the glider bounces off of the lower end of the track and may cross the timing gate a second time, thereby giving you erroneous readings. You may find it convenient to catch the glider after the first pass.
3. Select a point along the track to perform your experiment. Make measurements with the 30mm, 20mm, etc. flags. You probably want to do this several (maybe five) times with each flag to get an average t for each x. It is critical that you properly measure the flags!
4. Plot the average velocities (one for each flag) versus time interval.
5. The instantaneous velocity (the speed of the glider as the very front of the flag just reaches the detector) and the acceleration of gravity can estimated from the slope of the curve. Estimate a and a.
6. From this value of a and the angle of the air track you can determine and experimental value of g. Compare this with the true value.

DATA ANALYSIS:

1. Make up a table with columns to record pulses for the various sized flags. Allow conversion to time intervals.
2. Calculate average  t for each  x.
3. Calculate average velocities for each flag width.
4. Plot these values on a linear graph. Use error bars to show the uncertainties in velocity estimates.
5. Estimate instantaneous velocity. You will need to use Linear Regression on EXCEL to do this.
6. Report the instantaneous velocity.
7. Calculate the acceleration on the air track. If the above plot is a straight line, the acceleration is constant. The acceleration is the slope of this line. Determine the uncertainty in acceleration.
8. From the known acceleration a, calculate the acceleration due to gravity using the known slope of the air track.
9. Compute the uncertainty in this estimate.
10. Compare your value of g with the accepted value. Report the percent error.

NOTES:

1. Background light is critical in this experiment. Too much scattering into the detector may result in erroneous readings. Draw the shades in the laboratory to eliminate excessive light.
2. Use the linear regression program in EXCEL to develop the proper equation with slope and y-intercept.
3. Measurement of the width of the flags is critical. Measure them to ±.1 mm. Be sure to have the flag vertical and in the same position on the glider for each trial. You may wish to use a magnifying lens to help read these values precisely.

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