Dimensional Analysis
Part IV.
Combination Units
Dimensional Analysis IV.
Combination Units
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Typical problem:
The speed of a car is 60 miles/hour;
express this speed in km/min.
The speed of a car is 60 miles/hour;
express this speed in km/min.
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Steps to solve
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1. Identify starting units
2. Identify ending units
3. Identify unit conversion factors for both
the numerator and denominator of the unit
(you may need more than one)
4. Arrange conversion factors in an equation
so they cancel out all the starting units and
changes them into final units (No numbers)
5. Put in the numbers and solve the equation
The speed of a car is 60 miles/hour;
express this speed in km/min.

Step 1. Identify starting units
miles/hour
The speed of a car is 60 miles/hour;
express this speed in km/min.

Step 2. Identify ending units
km/min
The speed of a car is 60 miles/hour;
express this speed in km/min.

3. Identify unit conversion factors for both
the numerator and denominator of the unit
(you may need more than one)
End
Start
kilometers/minute
Miles/hour
Numerator
Miles to kilometers
1 kilometer = 0.62137 mile
Denominator
hours to minutes
1 hour = 60 minutes
The speed of a car is 60 miles/hour;
express this speed in km/min.
4.
Arrange conversion factors in an
equation so they cancel out all the
starting units and changes them into
final units (No numbers)
miles/hour x (km/mile) x (hour/minute)
Work on numerator and denominator
separately. Notice how the unit to cancel
the denominator has to be on the upper
part of the conversion fraction
The speed of a car is 60 miles/hour;
express this speed in km/min.
Step 5: Put in actual numbers and solve the problem
60 miles/hr x (1 km/0.62137 mi) x (1 hr/60 min)
= 1.609347 km/minute
=2 km/min (rounding to 1 significant figure)
Practice Problems
The next three pages have practice
problems, try out the problems and see if
you are doing them right
The speed of light is 2.998x108 m/s.
What is this in miles/hr
Step 1: Starting Units
meters and seconds
Step 2: Ending Unit
miles and hours
Step 3: Unit conversions
1 km = 0.62137 mi; 1 km = 1x103 m
60 seconds = 1 minute; 60 minutes = 1 hour
Step 4: Equation
m/s x (km/m) x (mi/km) x (s/min) x (min/hr)
Step 5: Solve
2.998x108 m/s x ( 1km/1x103m) x (0.62137 mi/1 km) x
(60 sec/minute) x (60 minutes/hour) =
6.706322136x108 miles/hour
6.706x108 miles/hour (rounded to 4 sig fig)
If I swim 2,100 meters/day, what is
this in miles/week
Step 1: Starting Units
m/day
Step 2: Ending Units
miles/week
Step 3: Conversion Factors
1 km = 0.62137 mile; 1 km = 1x103 m
7 days = 1 week
Step 4: Equation
meters/day x (km/m) x (mi/km) x (weeks/day)
Step 5: Solve
2,100 m/day x (1 km/1x103m) x (0.62137 mi/km) x
(7 days/week)
9.134 miles/week
=24 mi/week (2 sig fig)
Planck’s constant is 6.626x10-34 J·s;
Express this constant in btu·hour
Step 1: Starting Units
J·s
Step 2: Ending Units
btu · hour
Step 3: Conversion Units
1 btu = 1055.06 J
60 s = 1 min; 60 min = 1 hour
Step 4: Equation
J·s x (btu/J) x (s/min) x (min/hour)
Step 5: Solve
6.626x10-34 J·s x (1 btu/1055.06J) x (60 s/1 min) x
(60 min/1 hr)
= 2.260876x10-33 btu·hour
=2.261x10-33 btu·hour (4 sig fig)