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Motion in One Dimension

Introduction

The object of this lab is to help you gain understanding of motion in one dimension. We shall study the motion of a low friction air cart as it moves down a linear air cushioned track inclined at an angle θ. The motion along the track is one-dimensional and we shall choose the direction along the track to be the x-axis. The acceleration of the cart along the incline should be approximately constant and equal to g sin θ, where g = 9.81 m/s2 is the acceleration due to gravity. If we start the cart from rest at t = 0, and x = 0, then at time t, the speed vx and the position x of the cart are given by:
( 1 )
vx = at, and x =
1
2
at2 
By combining the two equations in Eq. (1)
vx = at, and x =
1
2
at2 
, we obtain the following relationship between vx and x.
( 2 )
vx2 = 2ax 
In these equations, a is the constant acceleration of the cart.

Procedure

In the first part of this experiment, we shall verify Eq. (1)
vx = at, and x =
1
2
at2 
by measuring the position of the cart as a function of time. In the second part of the experiment, we shall verify Eq. (2)
vx2 = 2ax 
by measuring the speed vx of the cart as a function of the position x.

Change of Position with Time

The experiment will use a very low friction air cart that runs on a cushion of air. This equipment is expensive, so please treat it well. You will use a stopwatch to measure the time it takes for the cart to reach a predetermined position. Note that the stopwatch you will be using is precise to 0.01 seconds, but your reflexes probably are not. Thus the measurements made here will not be highly accurate, and a careful error analysis will be needed to justify any conclusion that you make.
Figure 1

Figure 1

Velocity at Different Positions

One way to find the cart's velocity at position x is to simply time how long, Δt, the cart takes to go a small known distance, Δx, when it is passing the position x and use the equation vx = Δxt. You are still finding an average velocity by this method, but it is a good approximation of the cart's instantaneous velocity at x when Δt is small. The small time interval, Δt, can be measured precisely by using the lab counter/timer and the photogate.
Be sure that you and your TA each initial your worksheet.

Analysis

Position as a Function of Time

To check if the motion of the cart is described by Eq. (1)
vx = at, and x =
1
2
at2 
, you will make a plot of t2 vs. x (i.e. t2 on the y-axis and x on the x-axis). If indeed the acceleration is constant, the plot should produce a line described by:
( 3 )
t2 =
2x
a
 
Note: An alternate approach would be to plot x vs. t but this requires non-linear regression analysis.

Velocity as a Function of Position

To examine the validity of Eq. (2)
vx2 = 2ax 
, we will make a plot of vx2 vs. x.

Discussion

Summarize your results, i.e. your two values for the acceleration together with their uncertainty. Is the motion of the cart consistent with the equations of motion for constant acceleration Eq. (1)
vx = at, and x =
1
2
at2 
and Eq. (2)
vx2 = 2ax 
? Why or why not? Does the value of the acceleration found from the two different procedures agree with g sin θ? Explain. Which of the values is more precise? Which is more accurate? How can you tell? What are the most significant systematic errors in this experiment and how do they affect your results?