Class Notes Physics I, 8/31/98

Inferences from Initial and Final Velocities on a Known Time Interval, Constant Acceleration

Introduction, Goals and Questions

Determining Acceleration from Rest Given Displacement and Time Duration (unif. accel.)

position vs. Clock Time and Inferred Average Velocities for Flow from a Uniform Cylinder

Introduction, Goals and Questions

In a variety of experimental situations in this course, we will determine acceleration by timing an object as it accelerates from rest through a known displacement. We analyze this situation by reasoning it out, assuming a uniform acceleration. We then observe the position vs. clock time of the surface of a water column in a uniform cylinder from which water is flowing through a hole in the bottom of the cylinder, and from position vs. clock time data we infer velocity vs. clock time.

Determining Acceleration from Rest Given Displacement and Time Duration (unif. accel.)

If an object coasts 60 cm in 15 sec, starting from rest and accelerating at a uniform rate, then

• It has an average velocity of 4 cm / s.
• We conclude that its final velocity must be 8 cm / s.
• Its change in velocity must therefore be 8 cm / s so
• Its acceleration is 8 cm / s / (15 s ) = .53... cm / s.

This situation is depicted by the graph below.

http://youtu.be/qMiKEPeOgs8

• How does the graph make it clear that an average velocity of 4 cm / s, and initial velocity 0, imply final velocity 8 cm / s?

We determine for the above situation the displacement during each 5-second time interval, 

• We use ave. vel. * `dt (or trapezoid area) for each.
• We then infer position vs. clock time.

These calculations are represented in the figure below.

We note the following:

• A linear v vs. t graph always implies a parabolic s vs. t graph with position function s = a t^2 + bt + c.
• A linear v vs. t graph always implies constant acceleration.
• The constant acceleration is, somewhat confusingly, double the a of the quadratic position function.

We pose the following question:

• Why does a linear velocity vs. time graph give a curved position vs. time graph?

Position vs. Clock Time and Inferred Average Velocities  for Flow from a Uniform Cylinder

We observe the position s of the water surface vs. clock time t as water flows from a hole in the bottom of a uniform cylinder. 

We calculate time interval and displacement between each successive pair of data points, and use this to calculate the average velocity of the water surface over each time interval.  The time intervals `dt are indicated in the leftmost column, in red, and the displacements `ds in the rightmost column, also in red.

We associate the average velocity `ds / `dt on a time interval with the midpoint clock time (often indicated in these notes by tMid) on that interval.  The midpoint times are indicated in parentheses, in light blue, between the observed clock times.  Average velocities are indicated in the 'red' column inserted between the t and x columns.

Calculations are shown in the figure below.

We pose the following questions:

• Why do we associate the average velocity over an interval with the midpoint clock time for that interval?
• Is a graph of average velocity vs. midpoint clock time linear, or nearly so?
• Does the velocity of the water surface seem to change at an approximately constant rate?
• Could this be a uniform-acceleration situation?