Department of Chemistry and Physi 1201__Lab 5: Accelerated Motion__

**Name: **

**Preparation: **Read Classical Physics: Third Custom Edition for Mount Royal University, Sections 2.4 and 2.5

**Equipment:** Spark Table Data printout sheet, ruler, PhET simulation (for pre-lab questions).

**Learning goals:** To describe the uniformly accelerated motion of an object using position-time, velocity-time, and acceleration-time graphs and become acquainted with graphing techniques in Capstone analysis software.

__Introduction__

Acceleration is the rate at which the velocity changes. For example if we have two cars and we want them to achieve 50 km/h, the car that reaches this speed faster has a greater acceleration.

The acceleration of an object during the time interval Δt, in which the object’s velocity changes by Δ**v, **is the vector: **a**= Δ**v**/Δt

An object’s acceleration vector points in the same direction as the vector Δ**v**.

__Experiment__

In this lab, we will simulate accelerated motion using a puck launched on an air table. One end of the air table is elevated so that the table surface is tilted downwards. A paper is placed underneath the puck, on top of carbon paper. A spark timer leaves a series of dots on the paper as the puck moves at regular time intervals. These dots are used to investigate the motion of the puck. **In today’s lab, the spark timer was set to 50 ms to get more frequent data points than last lab.**

Figure 1. Position of the puck at 50 *ms* intervals

- Watch the video linked in Blackboard for on the Air Table in Accelerated Motion to observe the data collection.
- Use the printout of the spark timer data, titled Lab 5 Spark Table Accelerated Motion Data. You can find this in the Week 5 folder as well as in the package of essential printables. The beginning of the puck’s path is marked with a “O”, so that you know which way to orient the page.
- If you compare the beginning and end of the puck’s path, one end shows the dots more closely together and at the other end they are more spread out. Which phases of the puck’s motion correspond to these patterns?

Beginning of puck’s motion, shortly after its release:

End of puck’s motion, shortly before hitting the bottom of the table: - Explain why the dots created by the puck are closer together/further apart during these phases of motion:
- Choose a series of 12 consecutive data points that look like a
*reasonable*representation of the motion. Note that you are not obligated to choose the very first data point on your paper as point 0, and it is better to avoid doing so. If the puck was stationary when the very first data point was created, the time interval between the very first and the second data point may not be exactly 50*ms*. It is best to choose a region of 12 consecutive points that are at least 1 data point removed from the first dot. Choose points where the data looks well-behaved (not too much curvature). Label these points 0 to 12 on the paper. - Measure the
**position**of every data point (relative to data point 0,) on your sheet. Re-read the instructions from the Uniform Motion lab to remind yourself exactly how to do this. Record the position on the recording paper and in Table 1.**For now, fill in only the position column****and leave the uncertainty on the measurement blank. You will determine the uncertainty in upcoming questions.** - Analyze your instruments and the dataset (dots) and determine the
*instrumental**uncertainty*.

_______

- Analyze your instruments and the dataset (dots) and determine any
*observation**uncertainty*due to a curvature in the puck’s path. Re-read the instructions from the Uniform Motion lab to remind yourself exactly how to do this. Show your work below:

_______

- What is your estimation of the total absolute uncertainty for any given position measurement? Combine the two values of the two uncertainties as you did in the Uniform Motion Lab. Include this uncertainty in the position measurement in Table 1 and show your calculation below:

_______ - Include a scan of your data sheet with your markings at the end of this document in the Appendix.

Time( s) | Position( mm) | Displacement( mm) | Velocity( mm/s) | Acceleration( mm/s^{2}) | |

Table 1. Data representing the motion of a puck sliding down an inclined spark table. **Do not complete the displacement, velocity, or acceleration columns until the instructions tell you to do so.**** Only position**

- Use the position and time data in Table 1 to plot a graph by hand of position-versus-time using mm spaced graph paper. Be sure to label your axis, provide a title, and
**include error bars**(assume there is no uncertainty in time). The length of your error bars must match the actual size of the absolute uncertainty. Use “nice” numbers when choosing your axis scale, so that the smallest tick marks represent a multiple of 1, 2, 5, or 10.

- Based on the graph you plotted, what type of motion did the puck undergo? Explain what information you used to conclude this. And what kind of function does your graph resemble? If it resembles more than one type of function, use the equations for accelerated motion to justify the
*most likely*function. - Include your position vs time graph at the end of this document in the Appendix.

**Group Checkpoint ****– **You must complete up to this point *at a minimum* before meeting with your group members and instructor during the synchronous lab hour.

**Calculating Velocity**

This week your position vs time graph was not uniform. It should look more like the graph in Figure 2, where a best fit curve of the data is shown in black (it is up to you to describe what mathematical function this should represent). The velocity at a particular point is given by the slope of the tangent line to the curve at that point. This is illustrated by the green line, which runs tangent to the best fit curve at . The slope of the green line represents the velocity at .

Figure 2. Example of a position vs time graph of a puck undergoing uniform acceleration. The green line represents the tangent of the best fit curve at the point at . The slope of the green line gives the velocity of the puck at that instant in time.

One option to obtain the velocity at each point is to draw a best fit curve to fit the data, then draw a tangent line at regular time intervals, and then calculate the slope of each tangent line. As this is a little time consuming, we will use an approximation that works very well for data undergoing uniform acceleration: the displacement between two points can be used to approximate the average velocity at the midpoint in time between the two points. This is show in Figure 3, where the points at and are used to approximate the velocity at . Imagine a line drawn to connect the points at and , shown in Figure 3 in blue. The slope of this line can be calculated using the rise and run between the points:

This works out to a velocity (displacement over time). It turns out that the slope of the line connecting the two points is parallel to the tangent line at the midpoint between them (the green tangent line at ). You can observe this by comparing the slopes of the blue and green lines to see that they are parallel. Therefore, the velocity at can be calculated using the displacement between and ( on the graph) and the time difference between and (). As the slope is parallel to the slope of the tangent line at , it is a good approximation for , the velocity at time . This approximation will be used to calculate the velocity at the odd numbered time intervals (, , , etc.).

Figure 4. The slope of the line connecting the points at and , shown in blue, runs parallel to the slope of the tangent line at the midway between the two points at .

- To begin using this approximation, calculate the
**displacement**of the puck between even numbered data points and record the values in the third column of Table 1. Show a sample calculation below: - Calculate the uncertainty of the displacement by propagating the uncertainty of the two position measurements. Reminder of subtraction rule:

ifShow a sample calculation below:

- Use this approximation to calculate the velocity of the puck at odd numbered time intervals and record the values in the fourth column of Table 1. Show a sample calculation below:

- Use uncertainty propagation to determine the uncertainty in the velocity from the uncertainty in the displacement. Assume that there is no uncertainty in the time measurement.

Reminder of the division by a constant rule: if** **** ** then** **

when *c *is a constant value. Show a sample calculation of the uncertainty propagation:

- Use the velocity and time data in Table 1 to plot a graph by hand of velocity-versus-time using mm spaced graph paper. Be careful that the time you use for each datapoint corresponds to the actual time for that velocity value (hint: they should all be odd time intervals). Be sure to label your axis, provide a title, and
**include error bars**(assume there is no uncertainty in time). The length of your error bars must match the actual size of the absolute uncertainty.

- What function does this graph approximate? What does that tell you about the velocity?
- Create a best fit line in order to find the average slope of the graph. Calculate the slope, showing your work here:

m = ________________

- Create a best fit line in order to find the average slope of the graph. Calculate the slope, showing your work here:

**Appendix**

Include the following required scans here:

- Spark table data sheet with your labelling
- Position vs time graph
- Velocity vs time graph
- Acceleration vs time graph

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