## 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?

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?

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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