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 7.3 - Initial Conditions of Motion

The concepts of forces, accelerations, velocities and distance are not limited to free-fall motion. Constant accelerations exist in many other physical phenomena’s. Before ending we need to understand how we can modify our equations of motion to be consistent with any initial conditions that may exist. For example, a ball may be dropped with an initial velocity or a car may accelerate from a certain distance from a starting point. 

If an object was moving at constant velocity, , its velocity function would be:

From the definition of the derivative, the derivative of a constant function of the form is zero because:

Therefore the derivative of the velocity function is:

Looking back at our free-falling body, we know that its acceleration was . This can now be written as:

The anti-derivative of this acceleration function is then:

Remember that the anti-derivative of zero is a constant. At t=0 we have the initial condition:

We call , or the velocity of the object at t= 0. It represents a situation that may exist when the initial condition is zero. When anti-differentiating we need to remember to add a constant along to reflect the initial conditions that may exist in the situation. For example if a car is traveling down the highway at 82 mph , suddenly sees a cop, and then steps on the pedal, accelerating the cart at 3 mph/s, then its velocity at any time t, were t is measured as soon as he steps on the gas is:

We know that the velocity function of a free-falling body is:

The anti-derivative of this function gives us the distance covered as a function of time:

is the objects initial position at t=0. Our result can be generalized for initial acceleration or distance as follows:

Questions
  1. Consider the following three cases for a free-falling body:

    1 - A ball is dropped from rest from the top of a building.

    2 - A similar ball is dropped from the same spot with an initial velocity of 10 m/s

    Find the distance function (distance covered as a  function of time) for each case.

  2. Consider three cars that pass a certain starting point.

    Car 1 - starts from rest with a constant acceleration of 6 m/s/s

    Car 2 - Has an initial velocity of 50 km/hr and maintains this constant velocity     with no acceleration.

    Car 3 - Has an initial velocity of 20 km/hr and an acceleration of 4 km/hr/ sec

    Car 4 - starts 5 km in front of all the other cars with no initial velocity but an        acceleration of 7 km/hr/sec.

    Derive the distance function for each car with reference to the distance covered from the staring point.

    Graph the distance function for each car.

  3. Last, determine at which distance each car will pass each other. You can do this by either looking at where the graphs of the paths intersect or by setting the distance functions equal to each other and solving for time.

Solutions

  • d(t) = 4.9t^2
  • d(t) = 4.9t^2 + 10t
  • car1: d(t)=10.8t^2
  • car2: d(t)=50t
  • car3: d(t)=2t^2 + 20t
  • car4: d(t)=3.5t^2 + 5
  • car1 and car2 intersect at about 235 m at a time of about 4.8 seconds
  • car1 and car3 intersect at about 55 m at a time of about 2.4 seconds
  • car1 and car4 intersect at about 10 m at a time of about 1 second
  • car3 and car4 intersect at about 6 m at a time of about .5 seconds
  • car2 and car3 seem to never intersect, unless the intersection is at time much greater than 6 seconds

Next section -> Section 8.1 - Using the First Derivative

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