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Class Notes 8/24/98

Velocity and Acceleration



Introduction, Goals and Questions

Two of the most fundamental quantities we measure in physics are velocity and acceleration.  Here we begin to work out the meaning of these concepts within the familiar context of a ball rolling down an incline.  We work with the idea of a rate.  We develop ways of representing these ideas and meanings by graphs, and we design an experiment to verify the obvious fact that there is an increase in velocity as a ball rolls down an incline.

Today we

Our starting idea is that when a ball is rolled down an incline we expect it to speed up.  Our experience of inclines leads us to the following questions:

You will be asked to design and conduct an experiment using a timer and a ball on a ramp to


Ball on incline:  Average Velocity, Average Rate, Motion of Speedometer

http://youtu.be/YddWjY7aSNA

http://youtu.be/vSKw6KJhysQ

We can determine the average velocity of an object by determining the time required to travel through a known displacement.

We call this an average velocity because the velocity keeps changing as the ball rolls down the incline. 

The idea of average rate can be understood by analogy with the average rate at which we are paid.

We can test whether a ball traveling down a greater incline, starting from rest, does in fact experience a greater change in its velocity. 

We might imagine riding inside a ball rolling down an incline.  If we imagine that we have some sort of speedometer in the ball, we expect that the speedometer needle would move, perhaps at a constant rate and perhaps at either an increasing or a decreasing rate.

Graphs of velocity vs. clock time; acceleration

The graph below depicts velocity vs. clock time for a ball.  There are two possible curves on the graph.  The red curve shows a velocity which increases at a constant rate, while the red curve shows a velocity which increases faster and faster. 

We say that velocity is represented by the blue curve is increasing at an increasing rate. We might also think about what the curve might look like the velocity is increasing, but at a decreasing rate.

We previously saw that the average velocity was the average rate at which position changes.  We are now interested in the average rate at which the velocity changes

ph01.jpg (16868 bytes)     ph01a.gif (6049 bytes)

http://youtu.be/DTV5kwe2odc

http://youtu.be/7V0H9nAcnck

http://youtu.be/Fyr2bQGr_Dk

The average rate at which velocity changes is therefore `dv / `dt = (2 cm / sec) / (1.7 sec) = 1.2 cm / sec / sec (approx.  The 'approx. indicates that the 1.2 is an approximate value of 2 / 1.7.  You should be sure to check all calculations done in these notes.  Their accuracy is not guaranteed.).

For the straight-line graph of v vs. t, the slope is the same between any two points.   This means that the average acceleration between any two points is the same as between any two points.  We therefore conclude that acceleration is constant, and call this a constant-acceleration graph. 

The figure below depicts our analysis of the situation where a speedometer on a car (a pretty fast car, obviously) changes by 45 mi / hr in 2 seconds (visualize a speedometer doing that and imagine what it would feel like to be riding in that car).

ph02.jpg (15158 bytes)

http://youtu.be/UlOdP3yzBoo

Acceleration at an Instant

The interestingly colored figure below shows a velocity vs. clock time curve with some clock time, denoted t0, indicated on the t axis.  The goal is to find the acceleration at the instant t = t0.

Note error:  The first picture refers erroneously about finding the velocity at t0.  It should have said 'acceleration at t0'.

ph03.gif (10937 bytes)    ph03a.gif (11327 bytes)

http://youtu.be/krcl8IokZU4

We first note that the average acceleration between two points is the slope of the segment between the points. 

Experiment

Design and conduct an experiment using a timer and a ball on a constant-incline ramp to

Distance Learning Students

Do the following, and retain a copy for your own future reference.   Suggested titles:  'yymmdd'_experiment_**** and  'yymmdd'_questions_****, where 'yymmdd' consists of the last two digits of the year (yy) followed by the two-digit month mm (01, 02, ..., 12 for Jan thru Dec.) and dd the day of the month (01, 02, ... ).   If on a pre-Windows 95 system, try 'yymmddex' and 'yymmddqu'.


Questions for next time:

What does the graph of position vs. clock time look like for constant-acceleration motion?

How can we obtain a graph of velocity vs. clock time from a position vs. clock time graph?

How can we obtain a graph of position vs. clock time from a velocity vs. clock time graph?

How can we obtain a graph of acceleration vs. clock time from a velocity vs. clock time graph?

How can we obtain a graph of velocity vs. clock time from an acceleration vs. clock time graph   



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