Pptgalileo S Ramp Lab

PPTGalileo’s Ramp Lab

Developer Notes

• This is in a slightly different format than other activities
• Include diagram/photo of ramp set-up
• This may need more detail/clarity
• Not sure if the chart format is the best way
• I want to emphasize the importance of this activity/sequence/concepts….but I’m obviously having trouble

• Added intro for teachers
• Added WU question
• Broke into lessons
• Added prediction using formula
• Deleted calculating d/t2 (was confusing)
• Added sequence of derivation

Goals

• Students should understand acceleration
• Find equation: d=1/2at2

Concepts & Skills Introduced

Area

Concept

Time required

Several class periods

Warm-Up Questions

Draw pictures of 4 ramps on the board- one horizontal and the other three at increasing slopes. Ask- Which ramp would you use to get the most accurate data and why?

Presentation

We found that we could figure out how long it takes to catch a falling object by measuring distance. We would ultimately like to use the distance to find time. Then we investigated the relationship between distance and time (speed). Then we observed that falling objects do not fall at a constant speed. Let’s continue to investigate this!

A good introduction is to hand out stopwatches to the class and have them time you dropping a tennis ball (from standing on a desk). Ask the timers to report their times. The times will vary quite a bit. Try it again and ask for the times. They still should vary quite a bit. Ask the class how you can minimize the error in timing. Suggestions like dropping it from a higher distance or slowing down the ball should come out of the discussion. Discuss how Galileo wanted to make the same kind of investigations about falling objects and how he decided to slow down the object by using a ramp. Demonstrate this by holding a ramp at 90o and “rolling” a ball bearing down it (this is the same as free-fall). Then, incrementally lower the ramp to various angles and roll the ball bearing down it to show how this slows down the “fall.” For this activity we will be investigating the motion of falling objects by slowing them down on a ramp.

Set-up

• Prop up one end of the ramp about 2-3 cm (3/4”). Thin textbooks work well.
• Take distance measurements from the bottom of the ramp. Laying the meter stick on the ramp helps.
• Use a wooden block (a 2” x 2” works well) to stop the ball bearing.

Timing Technique

• Good timing technique is important for getting accurate, consistent data. Have the students practice a few times before actually recording data.
• One group member should hold the ball bearing at the proper distance with a pencil. The ball bearing should be released by moving the pencil away from the ball. The timer should react to the release.
• There is a tendency to anticipate at the end of timing (when the ball hits the block). However, the timer reacted (with a delay) to the release. To have the same offset at the beginning and end of timing, the timer should react both at the beginning and the end. For this lab, have the students turn their heads and react to the sound of the ball bearing hitting the block. This way they are reacting rather than anticipating.

The following is an outline of the sequence. After the initial data collection, there are several lessons of analyzing the data to explore the underlying concepts. Each lesson may take more or less than one class period depending on your class schedule and how the students are doing.

Lesson 1- Data collection

• Make a table of 6 columns and 11 rows
• Label the first column as distance (m)
• Label the next 4 columns for 4 samples of time (s).
• Label the next column for the averages of the times (s)

• The students will be making another table later on as they compare distance, time, speed, t2, etc. This initial table is for the data collection. The second table need not be cluttered with all the extra, “raw” data.

• Gather times for 5, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 cm. 4 samples each.
• Find the averages.

Lesson 2- Analyzing distance vs. time

• Make a new table with 10 columns and 13 rows
• The first two columns are distance and time.
• Include the distance 0.
• Copy the data from your previous table for d and tave

• 1 row for labels, 12 rows for 12 distances, including 0.
• For now, have the students label the first and second columns only.
• See sample data table below*

• Graph d vs. t.
• Use a separate sheet and scale your graph to use most of the sheet.
• Do your best-fit line in pencil.

• Compare the class graphs.
• This is not a straight line, but a curve. What is going on? What do we see? Discuss in small groups.
• Equal times but increasing distances, which means increasing speed.
• What is the slope of this curve? (speed) Just like electric cars.
• What does it mean that the slope changes? (the speed changes, and increases)
• Two reasons the line curves
  - the ball covers greater distances in the same amount of time, which means
  - the speed of the ball increases

*Sample Data Table

Lesson 3- Analyzing speed vs. time. Discussing vaveand vfinal

• Since we’re interested in speed (velocity) let’s add that to our analysis.
• Add vaveto the table.
• vave= dtot/ttot
• calculate vave for each distance interval

• Reminder: velocity is speed with direction. The direction here is not changing but the magnitude is.

• Graph vave vs. t.
• Leave 2x the distance on the y-axis.
• (Compare to the graph of d/t for electric cars).

• Should be a straight line!
• That means that the speed changes consistently, which you might expect because the ramp has the same tilt all the way.
• Need 2x y-axis to leave room for vf

• What is the instantaneous final speed compared to the average speed?
• Discuss in groups

• With a straight line, the average is (vf+vi)/2.
• In this case, vi is 0, so vf=2vave
• The speed it is going at the end is twice the average speed.

• Add vfto the table.
• Calculate vf for each distance interval
• Graph final speed vs. time
• Put it on the same graph with average speed.

• Should be a straight line, too!
• Speed changes consistently from vito vf.
• Before, we calculated the average speed for each distance interval. Since the initial speed was 0 for each interval, we can use 2vave to find the final speed the bb was going right before it hit the block at each interval.

• What does the straight line mean?
• With d/t, the slope is the speed.
• What do you call the slope of a graph of speed vs. time? Discuss in groups.

• The rate of change of speed is consistent with time.
• The slope of the line (k) is called acceleration.
• Acceleration is the change in velocity over time. ∆v/t
• Units: [m/s/s = m/s2]
• Plenty of homework available here! (See acceleration exercises.doc)

Lesson 4- Verifying constant acceleration; looking at the data in other ways

• If the rate of change of speed is consistent with time, let’s compare the two.
• Calculate vf/t for each distance interval
• Predict what it will be.
• Add it to the table.
• What do you see in the table?
• What is that number?

• This is a table of what you see in the graph.
• You should see a consistent number!
• This is the acceleration of the bb on the ramp.
• This is equal to the slope of the vf/t graph.
• Reminder: we are actually calculating (vf– vi)/t, but vi= 0.

• Interpolate for t = 0, 1, 2, 3, & 4 seconds from the original graph.
• Construct a table for this data
• Test your interpolated data with your ramps & bb. Are they similar?

• This is to get consistent intervals of time.
• Shows the use of graphs.
• Teaches interpolation.
• This is a good prediction and validation.

• From your interpolated data draw a picture of distance and time and how they relate (for an object with a constant acceleration).

• Suggestion:
  - Each line is 1 second.
  - Draw horizontal bars of how far the ball goes.
  - Should get a sideways bar graph of a parabola.
• What do you see?
• For each unit of time, there are greater distances.

Lesson 5- Relating d, t, & a. Deriving d = 1/2at2

• We’ve found that a=v/t, and the a on the ramp is constant.
• If v=d/t, then ad/t/t Can we simplify that?
• a d/t2.

• Units: [m/s2]
• d vs. t2

• We want to find an equation that relates d, t, a.
• Why? Velocity is hard to measure, but d t are easily measured and we’ve seen that a is constant (on the ramp and for falling things). This would be a handy relationship that we could use to calculate rxn time!
• We’re using the equations we know to make another one that is more efficient for certain calculations!

• Derive the equation for the students, making sure to take it step-by-step (see below)*.
• The derivation may be intimidating to students, but allowing them to see how all these equations “fit” together is worthwhile. It really illustrates the power of equations.

*Deriving the formula- (There are other sequences you may try that lead to the same result).

Start with the simple equations we know:

vave = (vf + vi)/2

vave = d/t (totals)

a= (vf – vi)/t

Let’s assume that vi= 0. So,

vave = vf/2

a= vf/t

Let’s solve vave = vf/2 for vf. We get,

vf = 2vave

Let’s substitute the above equation for vfin a = vf/t. We get,

a = 2vave/t

Let’s substitute vave = d/t in the above equation. We get,

a = 2(d/t)/t

Simplify and we get:

a= 2d/t2

(We can rearrange the above equation to obtain d=1/2at2. This equation will be handy to calculate our reaction times- once we find the acceleration of falling objects due to gravity).

Deriving equations is easy and fun!

• Using the formula

• Set up a ramp at a higher angle than before.
• Gather time data for 100 cm.
• Predict the time for a 50 cm run.
• Test your prediction- measure time for 50 cm.

• This is a nice summary and prediction. (Also, it’s a good embedded assessment).
• Once they have dt for 100 cm, they can calculate a.
• Using the calculated a, they can find the t for d = 50 cm.
• Take away their bbs before they predict!

• Predict the distance for 1/2 of your 100 cm time.
• Test it.

Lesson 6- Follow-up; prediction

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