Unlock the Secrets of Dimensional Analysis, Conversion Factors, and Derived Units

From the given information, speed and distance are given and time is required.

Note that the speed of the ball is given in miles per hour, but the distance from the pitcher's mound to home plate is in meters. This suggests that a conversion from miles to meters is needed.

\underline{Miles to Meters} Conversion Factor = \frac{1 6 0 9 . 3 4 m}{1 mi}

Multiplying the given speed by the conversion factor will convert the speed from miles per hour to meters per hour.

91 mi / h

Mult

Multiply by $1609.34 m / mi$

91 mi / h \cdot 1609.34 m / mi

Simplify

WriteAsFrac

Write as a fraction

\frac{9 1 mi}{1 h} \cdot \frac{1 6 0 9 . 3 4 m}{1 mi}

CrossCommonFac

Cross out common factors

\frac{9 1 mi}{1 h} \cdot \frac{1 6 0 9 . 3 4 m}{1 mi}

CancelCommonFac

Cancel out common factors

\frac{9 1}{1 h} \cdot \frac{1 6 0 9 . 3 4 m}{1}

Multiply

146449.94 m / h

Because the answer is required to be in seconds, a conversion from hours to seconds has to be made. To do so, the conversion factor needs to have a quantity in hours in the numerator and the equivalent quantity in seconds in the denominator.

\frac{meters}{hour} \cdot \frac{hour}{seconds} = \frac{meters}{second}

Using the fact that that

1

hour is equivalent to

3600

seconds, the conversion factor can be set.

\underline{Hours to Seconds} Conversion Factor = \frac{1 h}{3 6 0 0 s}

Having the conversion factor set, the conversion can be performed.

146449.94 m / h

Mult

Multiply by $\frac{1 h}{3 6 0 0 s}$

146449.94 m / h \cdot \frac{1 h}{3 6 0 0 s}

Simplify

WriteAsFrac

Write as a fraction

\frac{2 4 4 9 . 9 4 m}{1 h} \cdot \frac{1 h}{3 6 0 0 s}

CrossCommonFac

Cross out common factors

\frac{2 4 4 9 . 9 4 m}{1 h} \cdot \frac{1 h}{3 6 0 0 s}

CancelCommonFac

Cancel out common factors

\frac{1 4 6 4 4 9 . 9 4 m}{3 6 0 0 s}

CalcQuot

Calculate quotient

40.680538 \dots m / s

RoundDec

Round to $2$ decimal place(s)

40.68 m / s

When units of distance are divided by units of speed, the units of distance cancel out, giving a result in units of time.

\frac{meters}{\frac{meters}{second}} = second

Therefore, to compute the time it takes the ball to reach the home plate, the distance must be divided by the speed.

t = \frac{d}{r}

Substitute values and simplify

SubstituteII

$d = 14.8 m$ , $r = 40.68 m / s$

t = \frac{1 4 . 8 m}{4 0 . 6 8 m / s}

DivByFracD

$\frac{a}{b / c} = \frac{a \cdot c}{b}$

t = \frac{1 4 . 8 m \cdot s}{4 0 . 6 8 m}

CrossCommonFac

Cross out common factors

t = \frac{1 4 . 8 m \cdot s}{4 0 . 6 8 m}

CalcQuot

Calculate quotient

0.363815 \dots s

RoundDec

Round to $2$ decimal place(s)

t = 0.36 s

The ball reaches home plate in about

0.36

seconds, around one-third of a second. This is the same amount of time it takes to blink!

Quantity (Symbol)	Expression	Base Units	Unit Symbol (Name)
Area $(A)$	length $\times$ width	$m^{2}$	$m^{2}$
Volume $(V)$	area $\times$ height	$m^{3}$	$m^{3}$
Speed $(r)$	$\frac{distance}{time}$	$m / s$	$m / s$
Acceleration $(a)$	$\frac{speed}{time}$	$m / s^{2}$	$m / s^{2}$
Density $(ρ)$	$\frac{mass}{length ^{3}}$	$kg / m^{3}$	$kg / m^{3}$
Force $(F)$	mass $\times$ acceleration	$kg \cdot m / s^{2}$	N (newton)
Pressure $(P)$	$\frac{force}{area}$	$N / m^{2}$	Pa (pascal)
Energy $(E)$	force $\times$ distance	$N \cdot m$	J (joule)
Power $(P)$	$\frac{force \times distance}{time}$	$J / s$	W (watt)
Frequency $(F)$	$\frac{1}{time}$	$1 / s$	Hz (hertz)

Quantity (Symbol)

Expression

Base Units

Unit Symbol (Name)

Area

(A)

length

\times

width

m^{2}

m^{2}

Volume

(V)

area

\times

height

m^{3}

m^{3}

Speed

(r)

\frac{distance}{time}

m / s

m / s

Acceleration

(a)

\frac{speed}{time}

m / s^{2}

m / s^{2}

Density

(ρ)

\frac{mass}{length ^{3}}

kg / m^{3}

kg / m^{3}

Force

(F)

mass

\times

acceleration

kg \cdot m / s^{2}

N (newton)

Pressure

(P)

\frac{force}{area}

N / m^{2}

Pa (pascal)

Energy

(E)

force

\times

distance

N \cdot m

J (joule)

Power

(P)

\frac{force \times distance}{time}

J / s

W (watt)

Frequency

(F)

\frac{1}{time}

1 / s

Hz (hertz)

Start Time	End Time	Time Interval (minutes)	Distance Covered (meters)
$10 : 00 A . M .$	$10 : 10 A . M .$	$10$	$1500$
$10 : 10 A . M .$	$10 : 15 A . M .$	$5$	$500$
$10 : 15 A . M .$	$10 : 22 A . M .$	$7$	$1120$

Start Time

End Time

Time Interval (minutes)

Distance Covered (meters)

10 : 00 A . M .

10 : 10 A . M .

10

1500

10 : 10 A . M .

10 : 15 A . M .

5

500

10 : 15 A . M .

10 : 22 A . M .

7

1120

Time Interval (minutes)	Distance Covered (meters)	Speed (meters per minute)
$10$	$1500$	$\frac{1 5 0 0}{1 0} = 150$
$5$	$500$	$\frac{5 0 0}{5} = 100$
$7$	$1120$	$\frac{1 1 2 0}{7} = 160$

Time Interval (minutes)

Distance Covered (meters)

Speed (meters per minute)

10

1500

\frac{1 5 0 0}{1 0} = 150

5

500

\frac{5 0 0}{5} = 100

7

1120

\frac{1 1 2 0}{7} = 160

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Catch-Up and Review

Explore

Combining Base Units

Explore

Distance to the Lightning

Challenge

Voyager $1$ Arrival Time

Discussion

Combining Length and Time Units

Example

Distance Between Train Stations

Hint

Solution

Discussion

Converting Units of Measure

Example

Fastball Time to Plate

Hint

Solution

Discussion

Defining New Units

Concept

Derived Units

Example

Acceleration of a Formula $1$ Car

Hint

Solution

Example

Impact Force

Hint

Solution

Example

Time vs. Speed Graph

Answer

Hint

Solution

Closure

Voyager 1 Arrival Year

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