Understanding Calculus

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  Table of Contents

  Preface
  1. Why Study
  Calculus
  2. Numbers
  3. Functions
  4. The Derivative
  5. Differentiation
  6. Applications
  7. Free Falling
  Motion
  8. Understanding
  Derivative
  9. Derivative
  Approximations
  10. Integration
  Theory
  11. Understanding
  Integration
  12. Differentials

  Inverse Functions
  Exponents
  Exponential
  Functions
  Applications of
  Exponential
  Functions
  Sine and Cosine
  Function
  Sine Function
  Sine Function -
  Differentiation and
  Integration
  Oscillatory Motion
  Mean Value
  Theorem
  Taylor Series
  More Taylor Series
  Integration
  Techniques

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CHAPTER 7 - Study of Free Falling Bodies

Chapter 7 - Study of Free Falling Bodies

Section 7.1 - What is a Force?

An object in motion is characterized by a changing position as a function of time. The derivative of the distance function with respect to time gives us the velocity of the object as a function of time. Furthermore the derivative of the velocity function, gives us the acceleration of the object as a function of time. But what causes an object to move? To understand particle dynamics we need to first understand the concept of force.

Newton's first law of motion states that an object in motion will remain in motion until acted on by a force. This observation is one of most important ones ever made as it offers a way of defining what a force is. According to this law, an object traveling in space at 100,000 km/hr will remain at that velocity forever provided no force acts on it.

For example, if you were to throw a ball in space it would forever continue in the same direction along with the same velocity with which it left your hand. Here, space refers to anywhere that is free from the influence of any gravitational force, electro-magnetic force, air-resistance or any other forces. Once set in motion an object will continue with that same velocity forever.

Since no force is required to keep an object in motion, then a force can be defined as that which changes the velocity of the object. Thus force is a measure of a resistance to a change in motion. Since motion is characterized by aconstant velocity, then a change in motion results in a change in velocity. By definition, a change in velocity is an acceleration.

This simple, yet profound conclusion tells us that forces are defined by accelerations. The force required to accelerate an object is proportional to the magnitude of the acceleration. The mass of the object is also a factor since the greater the mass, the greater its resistance to motion. Observationshows that the resistance to a change in motion is directly dependent on the amount of matter being accelerated.

We can define a Newton as the force equipped accelerate a body of unit mass, 1 kg, Therefore to accelerate a body of mass, m, the required force would be m times a.

This is read as, the force required to accelerate a body is directly related to its mass and the magnitude ofthe acceleration of the mass. The important concept to understand is that forces are defined as accelerations or changes in velocity. It requires no force to keep a body in motion Once in motion it will remain in motion. A force is required only to change its velocityor accelerate it. Thus force is a quantifiable measurement of a massís resistance to a change in motion. If an object had no resistance to a change in motion then there would be no such thing as force!

This might seem to contradict reason. One can better understand this by considering an airplane flying in space, where space is some imaginary place that contains no matter or force fields inside it. If its four engines produce an acceleration and the mass of the plane is kg, then the thrust or force acting on the plane is . If we assume an inexhaustible and weightless fuel source then theoretically the engines will push the plane forward with a constant force of 4 million newtons.

Now since the plane is flying in an imaginary space under a constant force, it is free to accelerate forever. Remember forces are defined as accelerations and not velocities. The plane will accelerate at a constant acceleration of . This means the planes velocity would increases and increase at the rate of The graph of its velocity as a function of time would be a linearly increasing function:†††††

The derivative of the velocity function is the acceleration function:

The fundamental concept to understand here is that a force is required only to change an objects velocity. A change in velocity is by definition an acceleration. Therefore forces are required only to accelerate an object. A constant force acting on a body will accelerate the body with a constant acceleration, which means the bodyís velocity will increase and increase forever,all due to a constant force. Furthermore the greater the mass, the greater its resistance to a change in velocity. Thus, the force required to accelerate a mass is directly proportional to its mass.

Next section -> Section 7.2 - Understanding Free-Fall Motion

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