The study of the relationship between the motion of bodies or objects and the forces acting on them is often called the study of “force and motion.” In a more specific sense, the relationship between velocity, acceleration, and distance is known as kinematics.

## Uniform Motion

Motion may be defined as a continuing change of position or place, or as the process in which a body undergoes displacement. When an object is at different points in space at different times, that object is said to be in motion, and if the distance the object moves remains the same for a given period of time, the motion may be described as uniform. Thus, an object in uniform motion always has a constant speed.

## Speed and Velocity

In everyday conversation, speed and velocity are often used as if they mean the same thing. In physics, they have definite and distinct meanings. Speed refers to how fast an object is moving, or how far the object will travel in a specific time. The speed of an object tells nothing about the direction an object is moving. For example, if the information is supplied that an airplane leaves New York City and travels 8 hours at a speed of 150 mph, this information tells nothing about the direction in which the airplane is moving. At the end of 8 hours, it might be in Kansas City, or if it traveled in a circular route, it could be back in New York City.

Velocity is that quantity in physics which denotes both the speed of an object and the direction in which the object moves. Velocity can be defined as the rate of motion in a particular direction. Velocity is also described as being a vector quantity, a vector being a line of specific length, having an arrow on one end or the other. The length of the line indicates the number value and the arrow indicates the direction in which that number is acting.

Two velocity vectors, such as one representing the velocity of an airplane and one representing the velocity of the wind, can be added together in what is called vector analysis. Figure 1 demonstrates this, with vectors “A” and “B” representing the velocity of the airplane and the wind, and vector “C” being the resultant. With no wind, the speed and direction of the airplane would be that shown by vector “A.” When accounting for the wind direction and speed, the airplane ends up flying at the speed and direction shown by vector “C.”

Imagine that an airplane is flying in a circular pattern at a constant speed. The airplane is constantly changing direction because of the circular pattern, which means the airplane is constantly changing velocity. The reason for this is the fact that velocity includes direction.

To calculate the speed of an object, the distance it travels is divided by the elapsed time. If the distance is measured in miles and the time in hours, the units of speed will be miles per hour (mph). If the distance is measured in feet and the time in seconds, the units of speed will be feet per second (fps). To convert mph to fps, divide by 1.467. Velocity is calculated the same way, the only difference being it must be recalculated every time the direction changes.

## Acceleration

Acceleration is defined as the rate of change of velocity. If the velocity of an object is increased from 20 mph to 30 mph, the object has been accelerated. If the increase in velocity is 10 mph in 5 seconds, the rate of change in velocity is 10 mph in 5 seconds, or 2 mph per second. If this were multiplied by 1.467, it could also be expressed as an acceleration of 2.93 feet per second per second (fps/s). By comparison, the acceleration due to gravity is 32.2 fps/s.

To calculate acceleration, the following formula is used.

Acceleration (A) =

__Velocity Final (Vf) − Velocity Initial (Vi)__ Time (t)

Example: An Air Force F-15 fighter is cruising at 400 mph. The pilot advances the throttles to full afterburner and accelerates to 1,200 mph in 20 seconds. What is the average acceleration in mph/s and fps/s?

A =

__Vf____−____Vi__ t

A =

__1200____−____400__ 20

A = 40 mph⁄s, or multiplying by 1.467, 58.7 fps⁄s

In the example just shown, the acceleration was found to be 58.7 fps/s. Since 32.2 fps/s is equal to the acceleration due to gravity, divide the F-15’s acceleration by 32.2 to find out how many G forces the pilot is experiencing. In this case, it would be 1.82 Gs.

## Newton’s Law of Motion

*First Law*

When a magician snatches a tablecloth from a table and leaves a full setting of dishes undisturbed, he is not displaying a mystic art; he is actually demonstrating the principle of inertia. Inertia is responsible for the discomfort felt when an airplane is brought to a sudden halt in the parking area and the passengers are thrown forward in their seats. Inertia is a property of matter. This property of matter is described by Newton’s first law of motion, which states:

Objects at rest tend to remain at rest and objects in motion tend to remain in motion at the same speed and in the same direction, unless acted on by an external force.

*Second Law*

Bodies in motion have the property called momentum. A body that has great momentum has a strong tendency to remain in motion and is therefore hard to stop. For example, a train moving at even low velocity is difficult to stop because of its large mass. Newton’s second law applies to this property. It states:

When a force acts upon a body, the momentum of that body is changed. The rate of change of momentum is proportional to the applied force. Based on Newton’s second law, the formula for calculating thrust is derived, which states that force equals mass times acceleration (F = MA). Earlier in this site, it was determined that mass equals weight divided by gravity, and acceleration equals velocity final minus velocity initial divided by time. Putting all these concepts together, the formula for thrust is:

Force =

__Weight (Velocity final − Velocity initial)__ Gravity (Time)

Force =

__W (Vf − Vi)__ Gt

Example: A turbojet engine is moving 150 lb of air per second through the engine. The air enters going 100 fps and leaves going 1,200 fps. How much thrust, in pounds, is the engine creating?

F =

__W (Vf − Vi)__ Gt

F =

__150 (1200 − 100)__ 32.2(1)

F = 5,124 lb of thrust

*Third Law*

Newton’s third law of motion is often called the law of action and reaction. It states that for every action there is an equal and opposite reaction. This means that if a force is applied to an object, the object will supply a resistive force exactly equal to and in the opposite direction of the force applied. It is easy to see how this might apply to objects at rest. In application, as a man stands on the floor, the floor exerts a force against his feet exactly equal to his weight. This law is also applicable when a force is applied to an object in motion.

Forces always occur in pairs. The term “acting force” means the force one body exerts on a second body, and reacting force means the force the second body exerts on the first. When an aircraft propeller pushes a stream of air backward with a force of 500 lb, the air pushes the blades forward with a force of 500 lb. This forward force causes the aircraft to move forward. A turbofan engine exerts a force on the air entering the inlet duct, causing it to accelerate out the fan duct and the tailpipe. The air accelerating to the rear is the action, and the force inside the engine that makes it happen is the reaction, also called thrust.

## Circular Motion

Circular motion is the motion of an object along a curved path that has a constant radius. For example, if one end of a string is tied to an object and the other end is held in the hand, the object can be swung in a circle. The object is constantly deflected from a straight (linear) path by the pull exerted on the string, as shown in Figure 2. When the weight is at point A, due to inertia it wants to keep moving in a straight line and end up at point B. It is forced to move in a circular path and end up at point C because of the force being exerted on the string.

The string exerts a centripetal force on the object, and the object exerts an equal but opposite force on the string, obeying Newton’s third law of motion. The force that is equal to centripetal force, but acting in an opposite direction, is called centrifugal force.

Centripetal force is always directly proportional to the mass of the object in circular motion. Thus, if the mass of the object in Figure 2 is doubled, the pull on the string must be doubled to keep the object in its circular path, provided the speed of the object remains constant.

Centripetal force is inversely proportional to the radius of the circle in which an object travels. If the string in Figure 2 is shortened and the speed remains constant, the pull on the string must be increased since the radius is decreased, and the string must pull the object from its linear path more rapidly. Using the same reasoning, the pull on the string must be increased if the object is swung more rapidly in its orbit. Centripetal force is thus directly proportional to the square of the velocity of the object. The formula for centripetal force is:

Centripetal Force = Mass (Velocity2) ÷ Radius

For the formula above, mass would typically be converted to weight divided by gravity, velocity would be in feet per second, and the radius would be in feet.

Example: What would the centripetal force be if a 10-pound weight was moving in a 3-ft radius circular path at a velocity of 500 fps?

Centripetal Force = Mass (Velocity2) ÷ Radius

Centripetal Force = 10 (5002) ÷ 32.2 (3)

= 25,880 lb

In the condition identified in the example, the object acts like it weighs 2,588 times more than it actually does. It can also be said that the object is experiencing 2,588 Gs, or force of gravity. The fan blades in a large turbofan engine, when the engine is operating at maximum rpm, are experiencing many thousands of Gs for the same reason.

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