Greenwood press

EQUATIONS
23

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pressure increases, the volume of the gas will decrease, and vice versa. For
example, when diving under water, the amount of pressure in your ear sockets
will increase, causing the amount of space to decrease until your ears “pop.” The
amount of space in your lungs also decreases when you are underwater, making
it more difficult to breath when scuba diving. One way to visualize this effect is
to bring a closed plastic container of soda onto an airplane, and then notice the
change in its shape during takeoff and descent due to varying pressures in the
earth’s atmosphere at different altitudes. If temperature, t, and quantity of gas in
moles, n, vary, then the equation can be extended to the ideal gas law, which is
pv = nrt, where r is the universal gas constant equal to 0.082 (atm L)/(mol K).
The escape velocity of an object represents the speed at which it must travel
in order to escape the planet’s atmosphere. On earth, it is the speed at which a
rocket or shuttle needs in order to break the gravitational pull of the planet. The
equation that relates the escape velocity, 
v
e
, to the mass, M, and radius, R, of a
planet is approximately 
v
2
e
= (1.334 × 10
−10
)(M/R). The equation is based on 
finding the moment when the kinetic energy, 
0.5mv
2
e
, of the rocket exceeds its
potential energy that is influenced by the earth’s gravitational pull, 
GM m/R,
where G is a gravitational constant, 
6.67 × 10
−11
, and m is the mass of the rocket. 
Setting these two relationships equal to one another, 
0.5mv
2
e
= GM m/R, sets 
up a situation that determines the velocity at which the kinetic and potential
energy of the rocket are the same. An m on both sides of the equation cancels and 
the equation simplifies to 
v
2
e
= (1.334 × 10
−10
)(M/R). The mass of the earth 
is
5.98 ⋆ 10
24
kg, and has a radius of 6,378,000 m. This means that a rocket
needs to exceed 11,184 meters per second to fly into space. That is almost 25,000
miles per hour! 
Equations involving the sum of reciprocals exist in several applications. For
instance, the combined time to complete a job with two people, 
T
c
, can be deter-
mined by the equation 
1/T
1
+ 1/T
2
= 1/T
c
, where 
T
1
and
T
2
represent the time
it takes two different individuals to complete the job. This equation is based on
the equation P = RT, where P is the worker’s productivity, R is the worker’s rate,
and T is the worker’s time on the job. Since two workers complete the same job,
they will have the same productivity level. This means that the two workers’ pro-
ductivity can be represented by the equations 
P = R
1
T
1
and
P = R
2
T
2
. The
productivity for both workers is based on a combined rate and different time, rep-
resented with 
P = (R
1
+ R
2
)T
c
. Substituting 
R
1
=
P
T
1
and
R
2
=
P
T
2
makes the 
equation
P =

P
T
1
+
P
T
2
T
c
. Dividing both sides by 
T
c
and canceling the pro-
ductivity variable leaves the end result, 
1
T
1
+
1
T
2
=
1
T
c
.
Suppose an experienced landscaper can trim bushes at a certain house in 3
hours, and a novice takes 5 hours to complete the same job. Together, they will
take 1 hour, 52 minutes, and 30 seconds to complete the task, assuming that they
are working at the same productivity level (i.e., they are not distracting each
other’s performance by chatting). This result was determined by solving the
equation
1
3
+
1
5
=
1
T
c
. If both sides of the equation are multiplied by the product