Pptgalileo S Ramp Lab

PPTGalileo’s Ramp Lab

Developer Notes

This is in a 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

Version / Date / Who / Revisions
05 / 2003/08/13 / sc /

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

06 / 2003/09/16 / dk /

added exercises
formatted to match template (partially)
re-wrote section on dt2

Goals

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

Concepts & Skills Introduced

Area

Concept

Physics / Acceleration; a = (vf – vi)/t, a = ∆v/t
Physics / vave = (vf + vi)/2; vf = 2vave
Physics / d=at2/2
Science / Finding patterns

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, velocity). Then we observed that falling objects do not fall at a constant speed. Let’s continue the investigation!

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”). A 1" (nominal, actually 3/4") thick board or a thin textbook 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. (See the stopwatch activity.)
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 down the ramp, 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

Concept / Student Activity / Teacher’s notes / Formula
Gather data /

Make a table of 6 columns and 12 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)

One row for labels, 11 for distances.
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.

Gather 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

Concept / Student Activity / Teacher’s notes / Formula
Data analysis /

Make a new table with 6 columns and 13 rows
The first two columns are distance and average 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*

Data analysis-
Graphing d vs. t /

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 gets steeper? (the speed changes, and increases)
The line curves because
- the ball covers greater distances in the same amount of time, which means
- the speed of the ball increases

/ v=d/t

*Sample Data Table

distance / time / vav / vf / a
d (cm) / t (s) / d/t / 2vav / vf/t / t2 / d/t2
5
10
20

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

Concept / Activity / Teacher’s notes / Formula
Average speed /

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 speed is.

/ vave= dtot/ttot
Average speed changes consistently /

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

/ vave/t=k
vave=kt
(y=mx)
Final speed is double average speed /

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

With a straight line, the average is (vf+vi)/2. This is only valid when acceleration is constant.
In this case, vi is 0, so vf=2vave
The speed it is going at the end is twice the average speed.

/ vave=(vf+vI)/2
(for constant acceleration)
vf = 2vave (when vi=0)
Speed changes consistently /

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.

/ vf = 2vave
Acceleration /

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!

/ vf/t=k
vf=kt (y=mx)
k=a
a=∆v/t
a=(vf-vi)/t
∆v=at

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

Concept / Student Activity / Teacher’s Notes / Formula
Equal time, equal change in velocity /

If the rate of change of velocity 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- dt2

Concept / Student Activity / Teacher’s Notes / Formula
distance traveled is proportional to time squared /

We've seen that a = v/t is a constant in our graph.
But what is v? v = d/t.
Substitute d/t for v. What do you get?

The slope of v/t is a straight line, like d/t was for the electric cars.
v/t = d/t/t = d/t2

/ dt2
d/t2 = k

If v/t is a constant, what do you think d/t2 will be?
Calculate t2 and add it to your table.
Graph d vs. t2.

d/t2 will be a straight line
The graph of d vs. t really is a parabola.
The distance traveled is proportional to the time squared.

Lesson 5- Relating d, t, & a. Deriving d = (1/2)at2

Concept / Student Activity / Teacher’s Notes / Formula
Derive a = 2d/t2;
(or d = at2/2) /

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 reaction 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.

/ a = 2d/t2

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

Start with the simple equations we know:

Average velocity is total distance/total timevave = dtot/ttot

Average velocity is initial velocity plus final velocity over two.vave = (vf + vi)/2

Acceleration is final velocity minus initial velocity over time.a= (vf – vi)/t

Assume that initial velocity is zero (which it is on our ramps).vi= 0.

So,vave = vf/2

anda= vf/t

Solve vave = vf/2 for vf. vf = 2vave

Substitute the above equation for vfin a = vf/t. a = 2vave/t

Substitute vave = d/t in the above equation. a = 2(d/t)/t

Simplifya= 2d/t2

Other equations in the formula familyd = at2/2, t = (2d/a)

Deriving equations is easy and fun!

Now we have acceleration derived directly from distance and time, which we can measure easily. This equation is for an object with constant acceleration starting from rest. The formula shows what we have learned - acceleration is constant, and distance is proportional to time squared. 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.

Lesson 6- Follow-up; prediction

Concept / Student Activity / Teacher’s Notes / Formula
Using the formula to predict /

Set up a ramp at a higher angle than before.
Gather time for 100 cm only.

This is a nice summary and prediction.
Once they have d and t for 100 cm, the can calculate a.
Take away their bbs before they predict!

/ a= 2d/t2

Predict the time for 50 cm.
Test your prediction - measure time for 50 cm.

Using the calculated a, they can find the t for d = 50 cm.

/ t = (2d/a)

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

Using the calculated a, they can find the d for t/2.

/ d = at2/2

Background

Problem

Procedure

Summary

Reading

In this section of the course, we are looking at how things move (not why).

The first person to investigate things in a scientific way was Galileo Galilei (1564-1642). He is called the father of modern science because he insisted on testing ideas, not just thinking about them. Galileo was an amazing man. He discovered the principle that led to the invention of clocks by watching chandeliers swinging in church. He wondered about their frequency. But how to time them? He used his pulse. He discovered that the frequency of the chandelier was predictable. That information was later used to invent clocks, which in turn led to great advances in navigation and exploration. Galileo was also the first to explore space with a telescope and the first to see the moons of Jupiter. He was one of the first to say that the Earth was not the center of the universe but that it orbited the sun. That disagreed with the religion of the time, and he was put in jail for his scientific beliefs.

From the time of Aristotle (384-322 BC) until Galileo, most people believed that things fall at a constant speed and that heavier things fall faster. Galileo wanted to test if that was true. Other people had questioned it, but Galileo was the first to investigate it thoroughly. He had a couple of problems, however. Things fall fast, and clocks and watches hadn't been invented yet. He reasoned that objects going down a ramp accelerate in a similar way to falling things, but slower. That took care of the falling fast problem. But how to time them? He let water run into a container, then measured the amount of water. Although he didn't get exact times as we think of them, he could still compare the times.

Our experiment was similar to Galileo's, and we found the same things.

Idea / Formula
As things fall, their speed increases - they accelerate.
Equal time, increasing distance. / a = ∆v/t
The speed increases at a constant rate - they have constant acceleration.
Equal time, equal increase in speed.
(These equations are for an object starting from rest, vi = 0.) / a = vf/t
vf = at
The average speed is one-half the final speed. / vav = vf/2
The distance traveled is proportional to the time squared. / d = at2/2

By being scientific - testing ideas - Galileo (and you) proved that things do not fall at a constant speed, but they have a constant acceleration, and that the distance they travel is proportional to the time squared.

Exercises

In our experiments with balls rolling down ramps,
What is the independent variable?
What is the dependent variable?
Identify 3 controls.
In this graph,
what can you say about the change in distance per unit of time?
what can you say about the velocity?
Discuss the slope of the line and what it means.
In this graph,
what can you say about the change in distance per unit of time?
what can you say about the velocity?
Discuss the slope of the line and what it means.

In this graph,
what can you say about change in velocity per unit of time?
What is the change in velocity per unit of time called?
Discuss the slope of the line and what it means.

In this graph,
what can you say about the change in acceleration per unit of time?
what can you say about the velocity?
Discuss the slope of the line and what it means.
Can an object be moving if its acceleration is zero?
As an object falls, what happens to its speed? How do you know?
If you wanted to drive a stake into the ground by dropping a rock on it, would you drop the rock from a low height or a higher height? Why?
Why does it hurt more when you land if you jump from a higher place?
In football, players fall to the ground or get driven to the ground many times and (most of the time) they don't get hurt. In the stands, there are railings to keep people from falling. What's the difference?
On which of these hills would a ball roll down with increasing speed and decreasing acceleration?

Why does a stream of water get narrower as it falls from a faucet?
What is the acceleration of a car moving at a constant velocity of 27 m/s for 100 s?
Which is greater, an acceleration from 25 km/h to 30 km/h, or an acceleration from 96 km/h to 100 km/h, if they both happen in the same amount of time?
What is the average velocity of a vehicle that accelerates constantly from 20 km/hr to 50 km/hr?
If a car accelerates at a constant rate from a traffic light, and its average speed is 20 km/hr, what is its final velocity?
Do policemen care about your final velocity, your average velocity, or your instantaneous velocity?
If a ball rolling down a ramp accelerates at a constant 3 m/s2, what is its acceleration after 4 seconds?
A ball rolling down a straight ramp, starting from rest, goes 1 m in 5 seconds.
What is its average velocity?
What is its final velocity?
What is its acceleration?
If a car is stopped at a stoplight, the light turns green, and 8 seconds later its speedometer reads 20 m/s, what is its average acceleration?
If a car can accelerate at 8 m/s every second, how long will it take to go from 0 m/s to 32 m/s?
If a ball rolling down a ramp covers 80 cm in 2 s, what is its acceleration?
If a ball bearing rolling down a ramp accelerates at 0.1 m/s2,
how fast will it be going at the end of 2 seconds? (What is its final velocity?)
What will its average speed be?
How far will it have gone?
A couple on a tandem bicycle accelerates from rest at a constant 1 m/s2. How far will they travel in 5 seconds?
If a falling object accelerates from rest at a constant 10 m/s2, how long will it take to go 45 meters?
A car is going 20 m/s, and it decelerates at 5 m/s2. How far will it travel before it stops?
What is the acceleration of an object that starts from rest and goes 27 cm in 3 seconds?
Fill in the table for an object with constant acceleration.

Time (s) / Distance (m) / Acceleration (m/s2)
0 / 0
1 / 2
2
3

What is the average velocity for a car which leaves school at 3:35 exactly, accelerates to 30 mph in 10 s, stays at 30 mph for 2 minutes, decelerates to a stop at a red light in 5 s, accelerates to 50 mph on the highway in 15 s, stays at 50 mph on the highway, decelerates on the off ramp to 25 mph in 5 s, stays at 25 mph for 5 minutes, then decelerates to a stop in the garage in 8 s, arriving at 3:57 exactly, having traveled 11 miles?
If an object has a constant acceleration from rest, and it goes some distance in an amount of time, how far will it go in twice as much time?
If an object has a constant acceleration from rest, and it takes some amount of time to go a certain distance, how much time will it take to go nine times as far?
For an object that is accelerating at a constant rate, draw the following graphs (no units needed)
d on the vertical axis and t on the horizontal axis
v on the vertical axis and t on the horizontal axis
a on the vertical axis and t on the horizontal axis
You’re at the top of a cliff, and you wonder how high it is, so you drop a rock off. The rock takes 5 s to hit the bottom. About how tall is the cliff?
If you dive off of a 10 m platform, about how long will it take for you to hit the water?

Challenge/ extension