Kinematics of Constant Acceleration

Purpose

The purpose of this investigation is to determine the acceleration of a lab cart and verify the equations for constant acceleration.

Procedure

A spark timer will be used to measure the motion of a lab cart that rolls down a ramp and coasts to a stop on a level surface.  The ramp and level surface will be a track and a board.  The track at one end rests on the lab table and at the other end is supported by a ring stand in the sink.  See the diagram below.  Test the set up by releasing the cart from about halfway up the ramp and make sure that it will slow down and stop before rolling off the board.  Do not let it fall off the table!  Start it from a lower position if it coasts too far.

Place a spark timer at the top of the ramp and set it for 10 Hz operation.  Prepare a strip of timer paper long enough to measure the entire motion of the cart.  Feed the paper strip through the timer in the direction indicated, making sure that the metallic treated side of the timer paper is facing the top of the timer.  Tape the timer strip to the lab cart.  Place the cart about half way up the ramp.  Start the timer and release the cart.  The cart should stop before the end of the paper comes through the timer.  If you are successful you should now have a strip showing the acceleration and deceleration of the cart from the point where it was released to the point where it eventually came to a stop.  Mark the starting end of the strip so that you know which way the cart was moving.

Analyzing the Strip

Locate the first ten pairs of dots that represent the cart’s rolling down the ramp.  Label the pairs A, B, C, … through J.  Measure the total distance of each point from point A – point A is the initial point of rest.  This is the magnitude of the position, x, measured from point A.  Measure to the nearest 0.01 cm.  Complete the data table as shown in the example below.  Note how the table cascades; the shaded cells are left blank. 

Point	t (s)	x (cm)	Δx (cm)	v (cm/s)
A	0.00	0.00
	0.05		0.26	2.6
B	0.10	0.26
	0.15		0.35	3.5
C	0.20	0.61
	0.25		0.47	4.7
D	0.30	1.08

Time, t, is the total elapsed time from the initial point A.

Position, x, is measured from the point of reference, point A.

Displacement, Δx, is the change in position values between the labeled points.

Velocity, v, is change in position divided by change in time.

Repeat the process for ten pairs of dots during which the cart slows but does not stop.  These points will be located on the part of the strip where the cart was coasting on the level board and slowing down.  It does not matter which ten pairs of dots but do not go all the way to the end (somewhere “in the middle” would be best).  Note that in this set of data the time “starts over” at zero as does the position.

Finally count the number of pairs of dots from one end to the other in order to determine the total time that the cart was in motion.  Also measure the total distance traveled by the cart from its release to its eventual stop.  Record these results in the table.

Interpretations

1. Make a velocity vs. time graph for the cart rolling down the ramp.  Determine the best fit and equation.  Always show all work and results on the graph.

2. Make a position vs. time graph for the cart rolling down the ramp.  Determine the best fit and equation.

3. Make a position vs. time squared graph – include a small table showing these values.  Determine the best fit and equation.

4. Make a velocity vs. time graph for the cart coasting (and slowing down) on a level surface.  Determine the best fit and equation.

5. Make a position vs. time graph for the cart coasting on a level surface.  Instead of finding the best fit, you will use the results of the previous graph to plot a curve.  Identify the initial velocity and acceleration values in the equation found on the velocity graph.  Plot an appropriate position function based on these values and connect with a smooth curve so that this “theoretical” relation can be compared to the data.  Show your work right on the graph.  This theoretical relation is one of the constant acceleration equations.  (This should be a very good fit but might not be the best fit.)

Questions

1. Explain whether or not the acceleration of the cart was constant during each interval of motion.  Refer specifically to the graphs and curve fitting results to support your response.

2. (a) Look at the equations on the graphs and compare the coefficient(s) from graph #2 and the coefficient(s) from graph #3.  Should these values be the same?  Explain.  (b) Determine the percent difference in the coefficients.

3. Discuss how well (or not) your results in this experiment support the equations of the standard model of constant acceleration.  Make specific references to support your answer.

4. (a) Use the total time that the cart was in motion and the values for its acceleration and deceleration to solve for the cart’s maximum speed.  (b) Make a careful sketch of the cart’s speed vs. time for its entire trip.  (c) Use this graph and the maximum to speed calculate the total distance traveled.  (d) Compare to the measured total distance and find the percent error in the calculated value (assuming the measured value is correct). 

5. Write a concise paragraph or two, free from errors in grammar and spelling, in which you discuss error in this experiment.  Any complete discussion of error will address the errors that are apparent in your results and it will address the most probable causes of those apparent errors.  In other words you should describe how your results are less than perfect and attempt to reasonably explain why the results are less than perfect.  Remember to consider both random and systematic error.

Your report (50 pts.) shall consist of the following material – neatly labeled and in this order:

q Completed data tables                                                                                      (5)

q 5 graphs as described in the interpretation section (and in that order)                  (35)

q Answers to questions 1 – 5                                                                               (10)

Cart Rolling Down Ramp
Point	t (s)	x (cm)	Δx (cm)	v (cm/s)
A	0.00
	0.05
B	0.10
	0.15
C	0.20
	0.25
D	0.30
	0.35
E	0.40
	0.45
F	0.50
	0.55
G	0.60
	0.65
H	0.70
	0.75
I	0.80
	0.85
J	0.90

Cart Coasting on Level Surface
Point	t (s)	x (cm)	Δx (cm)	v (cm/s)
A	0.00
	0.05
B	0.10
	0.15
C	0.20
	0.25
D	0.30
	0.35
E	0.40
	0.45
F	0.50
	0.55
G	0.60
	0.65
H	0.70
	0.75
I	0.80
	0.85
J	0.90

Entire Trip of Cart

total time

total distance