HPP A6
Estimating Instantaneous Velocity and Acceleration from Data
Our goal in this activity is to estimate the velocity-time data by doing a calculation involving the
postion data and to estimate acceleration-time data from velocity data.
Estimating Instantaneous Velocity and Acceleration
Exploration
The following table shows some position-time data for an object moving in a straight line.
t [s]
x [m]
0.00
0.50
0.25
0.63
0.50
1.00
0.75
1.63
1.00
2.50
1.25
3.63
1.50
5.00
1.75
6.63
2.00
8.50
2.25
10.63
GE 1.
1. What is the physics definition for average velocity over a specified time
period?
2. For the data in the table above, calculate the average velocity from 0 to 2
[s].
3. Calculate the average velocity from 1.75 to 2.25 [s].
Activity Guide
© 2010 The Humanized Physics Project
Supported in part by NSF-CCLI Program under grants DUE #00-88712 and DUE #00-88780
HPP A6
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4. Which value, from question 2 or 3, do you think would be a better estimate
for the velocity right at 2 [s]?
Invention
We define instantaneous velocity at a time t as the average velocity in a very small time interval
that surrounds t. Ideally, we would take a mathematical limit:
v = lim
Δt →0
Δx
Δt
where the Δt surrounds the time of interest. When dealing with real experimental data, taking
the limit is not really possible, so we estimate it by just calculating the average velocity over a
small time interval surrounding the time of interest.
The instantaneous acceleration can be defined in a similar fashion. We calculate the average
acceleration over a small time interval surrounding the time of interest and take a limit.
a = lim
Δt →0
Δv
Δt
Again, when dealing with real experimental data, taking the limit is usually not possible, so we
estimate it by calculating the average acceleration over a small time interval.
Application
We will apply these ideas to the data presented above. In the process, we will see how to get
Excel to do the calculations for us.
The first step is to add columns for calculating the velocity and acceleration, and then eliminate
every other position measurement, as shown below.
Activity Guide
© 2010 The Humanized Physics Project
HPP A6
3
t [s]
x [m]
0.00
V [m/s]
A [m/s2]
0.50
0.25
0.50
1.00
0.75
1.00
2.50
1.25
1.50
5.00
1.75
2.00
8.50
Next, we calculate average velocity over small time periods to estimate instantaneous velocity.
The velocity estimate at 0.25 [s] is
v≅
Δx (1.00 − 0.50 )
=
= 1.0 [m/s]
Δt (0.50 − 0.00)
Fill in the estimates for instantaneous velocity in the table below in the white cells. The first two
are done for you.
t [s]
x [m]
0.00
1.0
1.00
0.75
1.00
A [m/s2]
0.50
0.25
0.50
V [m/s]
4.0
3.0
2.50
1.25
1.50
5.00
1.75
2.00
8.50
Estimates for instantaneous acceleration are done similarly. For example, the acceleration at
0.50 [s] is
(3.0 − 1.0) = 4.0 [m/s 2 ]
Δv
=
Δt (0.75 − 0.25)
Fill in the remainder of the acceleration estimates.
a≅
Activity Guide
© 2010 The Humanized Physics Project
HPP A6
4
Using Excel
The final step is to get Excel to do all the calculations.
ƒ
Start Excel.
ƒ
Copy the time and position columns onto the clipboard.
ƒ
Go to the spreadsheet and click in cell A1.
ƒ
Paste the data into Excel.
Now we will construct an Excel formula for calculating average velocity over a small time
interval. Excel formulae always begin with an equal sign, "=".
ƒ
Select cell C3 and type in the formula: =(B4-B2)/(A4-A2)
ƒ
Push Enter.
Notice how the cell with the formula turns into a number. Excel takes the numbers in cells B4,
B2, A4, and A2 and does the calculation with them.
ƒ
Copy the formula onto the clipboard and paste into the appropriate cells in column C.
Excel should have calculated velocity values that equal the ones you calculated earlier. If this is
not the case, you should get the instructor.
Now construct an appropriate formula for estimating instantaneous acceleration at 0.50 [s] in cell
D4. Copy your formula into the appropriate cells in column D.
You should now have a spreadsheet that reproduces the table of calculations you did earlier.
Paste it here. You will need to do a Paste Special and select Microsoft Exce Worksheet Object.
Activity Guide
© 2010 The Humanized Physics Project