Examples of How to Use
the Most Important Equation in Physics
Example (1): Gravity at the Surface of the Earth
Using the universal law
of gravity (a subject that goes beyond
the present discussion), one can show that the Earth exerts a force of almost
200 Newtons on a rock with a mass of 20 kilograms (about 44 pounds). A Newton
is a unit of force equal to
kilograms meters/second2. A meter
is a little longer than a yard, while a kilogram is a little more
than 2 pounds. If the
rock is thrown off a cliff, how fast does it accelerate to the ground?
Answer:
Use Newton's second law a = F/m to find
a = (200 Newtons)/(20 kilograms)
= 10 Newtons/kilograms =
10 (kilograms meters/second2)/kilograms
= 10 meters/second2
Further Discussion:
When an object moves
in a straight line, such as a rock falling
vertically, the acceleration is the change in speed
per unit time. Hence, the little computation above tells us
that the rock increases its speed
at 10 meters/second each second. If the rock is initially released without
being thrown, its speed starts out as zero. After one second, its speed
becomes 10 meters per second, after two seconds its speed
reaches 20 meters per second, after three seconds it
is 30 meters per second, and so on. The rock is accelerating
at 10 meters/second2 since it is increasing
its speed by 10 meters per second each second.
One can also determine how far such a rock
falls. The distance an object travels is the average
speed v times the time of motion t. At one second, the
rock's speed is
10 meters/second. Initially, its speed was zero. So during
the first second, the rock's average
speed is 5 meters/second (the average of 0 meters/second and 10 meters/second).
Hence, after one second, the rock falls a distance d of
d = v t
(1)
= (5 meters/second) (1 second) = 5 meters .
How fast does it travel during
the first two seconds? The rock's speed at
two seconds is 20 meters/second while its initial speed was 0 meters/second. The
average of these two speeds is 10 meters/second. So in two seconds, the rock drops
d = v t
= (10 meters/second) (2 seconds) = 20 meters .
For the case of three seconds, the
rock's average speed is 15 meters/second (the average
of 0 meters/second and 30 meters/second). So in three seconds, it falls
d = v t
= (15 meters/second) (3 seconds) = 45 meters .
Note that the rock is falling
by successively larger amounts because the speed is continually
increasing. The above computations of the distance that the rock falls can be
summarized by the formula
d = 5 (meters/second2)
t2
(2)
and, indeed, this equation provides the result for any time t as long as
air friction can be neglected.
It turns out that when air friction
is negligible, all bodies fall with the
same acceleration at the surface of the Earth. This acceleration, which is
almost 10 meters/second2, is usually
denoted by the symbol g. Since a meter is a little more than
three feet, g also
equals 32 feet/second2. If
a penny is tossed off the cliff, it too would
fall 5 meters, 20 meters and 45 meters
after respectively one, two and three seconds. The formula in Equation (2)
can be written as d = gt2/2
since g = 10 meters/second2. In general when
an object begins at rest and undergoes constant acceleration of a, it travels
a distance d of d = at2/2 in time t.
Example (2): Car Manufacturing Requirements
Car producers, particularly those
of sports cars, often boast of how well their vehicles can achieve high
speed in short times. For example, the 1999
Ferrari 360 Modena 
can
go "from 0 to 60 in four seconds." Note that the last part of
this sentence means that the car can achieve a speed of 60 miles per hour
starting from rest. This is a statement about an acceleration: how
fast the car can change its speed. The Ferrari 360 Modena
weighs 1300 kilograms (and costs about $150,000!). How
much force must the V8 engines of the 360 Modena supply?
Answer:
Use Newton's second law F = ma. Acceleration is the change in speed
per unit time. The Ferrari changes its speed by 60 miles per hour in four
seconds so that its acceleration is (60 miles/hour)/(4 seconds) =
15 miles/hour/second. In each second, it can increase
its speed by 15 miles per hour. Now use
Newton's second law F = ma with m = 1300 kilograms to obtain
F = ma = (15 miles/hour/second) (1300 kilograms)
= 19500 kilograms miles/hour/second
Further Discussion:
The units are mixed in the above calculation. To
convert to Newtons, one needs to convert hours to seconds and miles to meters. One
mile is about 1610 meters and one hour is 3600 seconds. So
F = 19500 kilograms miles/hour/second =
19500 kilograms (1610 meters)/(3600 seconds)/second = 8700 Newtons
This force is more than 40 times the force that the rock in example (1) feels
due to the gravity of Earth.
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