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LESSON 5 : FREE FALL AND THE ACCELERATION OF GRAVITY

By Clyde Mills,2014-07-08 10:36
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A FREE-FALLING OBJECT IS AN OBJECT WHICH IS FALLING UNDER THE SOLE INFLUENCE OF GRAVITY. THUS, ANY OBJECT WHICH IS MOVING AND BEING ACTED UPON ONLY BY THE FORCE OF ...

Lesson 5 : Free Fall and the Acceleration of Gravity

Introduction to Free Fall

    A free-falling object is an object which is falling under the sole

    influence of gravity. Thus, any object which is moving and being acted

    upon only by the force of gravity is said to be "in a state of free fall." This definition of free fall leads to two important characteristics about a free-falling object:

    ; Free-falling objects do not encounter air resistance.

    ; All free-falling objects (on Earth) accelerate downwards at a

    rate of approximately 10 m/s/s (to be exact, 9.8 m/s/s).

    Because free-falling objects are accelerating downwards at a rate of

    10 m/s/s (9.8 m/s/s to be more accurate), a ticker tape trace of its

    motion depicts acceleration.

    The diagram at the right shows such a ticker tape trace. The position of the free-falling object at regular time intervals, every 0.1 second, is shown. The fact that the distance which the ball travels every interval of time is increasing is a sure sign that the ball is speeding up as it falls downward. If an object travels downward and speeds up as it travels down, then its acceleration is directed downward.

    This free-fall acceleration can also be demonstrated using a strobe light and a stream of dripping water. If water dripping from a medicine dropper is illuminated with a strobe light and the strobe light is adjusted such that the stream of water is illuminated at a regular rate say every 0.2 seconds; instead of seeing a stream of water free-falling from the medicine dropper, you will see several consecutive drops. These drops will not be equally spaced apart; instead the spacing increases with the time of fall (as shown in the diagram above), a fact which serves to illustrate the nature of free-fall acceleration.

    The Acceleration of Gravity

    A free-falling object is an object which is falling under the sole influence of gravity; such an object has an acceleration on Earth of 9.8 m/s/s, downward. This numerical value for the acceleration of a free-falling object is such an important value that it has been given a special name. It is known as the acceleration of gravity the acceleration for any object

    moving under the sole influence of gravity. As a matter of fact, this quantity known as the acceleration of gravity is such an important quantity that physicists have a special symbol to denote it the symbol g. The numerical value for the acceleration of gravity is most accurately known as 9.8 m/s/s.

    There are slight variations in this numerical value (to the second decimal place) which are dependent primarily upon altitude. Sometimes we will use the approximated value of 10 m/s/s in order to reduce the complexity of the many mathematical tasks performed with this number. By so doing, you will be able to better focus on the conceptual nature of physics without sacrificing too much in the way of numerical accuracy. When the moment arises that you need to be accurate (such as in lab work), use the more accurate value of 9.8 m/s/s.

    g = 10 m/s/s, downward

Recall that acceleration is the rate at which an object changes its velocity. Between any two

    points in an object's path, acceleration is the ratio of velocity change to the time taken to

    make that change. To accelerate at 10 m/s/s means to change your velocity by 10 m/s each

    second.

    The triangle is greek letter delta, which means “change in.” This is the same thing as writing Vf Vi as we had been before.

    If the velocity and time for a free-falling object being dropped from a position of rest were

    tabulated, you would notice the following pattern:

    Time (s) Velocity (m/s)**

    0 0

    1 10

    2 20

    3 30

    4 40

    5 50

    (**velocity values are based on using the approximated

    value of 10 m/s/s for g)

    The velocity-time data above reveals that the object's

    velocity is changing by 10 m/s each consecutive second.

    That is, the free-falling object has an acceleration of 10 m/s/s.

    Another way to represent this acceleration of 10 m/s/s is to add numbers to the ticker tape diagram from the first

    section of this packet.

    Assuming that the position of a free-falling ball dropped

    from a position of rest is shown every 1 second, the velocity of the ball will be shown to increase as depicted in the

    diagram at the right. (NOTE: This diagram is not drawn to

    scale it would take no more than two seconds for a ball to

    drop from shoulder height to toe height.)

Representing Free Fall by Graphs

    Position vs. Time Graphs

    The position vs. time graph for a free-falling object is shown below.

Observe that the line on the graph is curved. A curved line on a position vs. time graph

    signifies an accelerated motion. Since a free-falling object is undergoing an acceleration of g = 10 m/s/s (approximate value), you would expect that its position-time graph would be curved. A closer look at the position-time graph reveals that the object starts with a small velocity (slow) and finishes with a large velocity (fast). Since the slope of any position vs. time graph is the velocity of the object (Lesson 3), the initial small slope indicates a small

    initial velocity and the final large slope indicates a large final velocity. Last, but not least, the negative slope of the line indicates a negative (i.e., downward) velocity.

Velocity vs. Time Graphs

    The velocity vs. time graph for a free-falling object is shown below.

    Observe that the line on the graph is a straight, diagonal line.

    A diagonal line on a velocity vs. time graph signifies an

    accelerated motion. Since a free-falling object is undergoing

    an acceleration of g = 10 m/s/s (approximate value), you

    would expect that its velocity-time graph would be diagonal.

    A closer look at the velocity-time graph reveals that the

    object starts with a zero velocity (starts from rest) and

    finishes with a large, negative velocity; that is, the object is

    moving in the negative direction and speeding up. An object

    which is moving in the negative direction and speeding up is

    said to have a negative acceleration Since the slope of any velocity vs. time graph is the acceleration of the object, the constant, negative slope indicates a constant, negative acceleration. This analysis of the slope on the graph is consistent with the motion of a free-falling object an object moving with a constant acceleration of 10 m/s/s in the downward direction.

How Fast? and How Far?

    Free-falling objects are in a state of acceleration. Specifically, they are accelerating at a rate

    of 10 m/s/s. This is to say that the velocity of a free-falling object is changing by 10 m/s every second. If dropped from a position of rest, the object will be traveling 10 m/s at the end of the first second, 20 m/s at the end of the second second, 30 m/s at the end of the third second, etc.

    The velocity of a free-falling object which has been dropped from a position of rest is dependent upon the length of time for which it has fallen. The formula for determining the velocity of a falling object after a time of t seconds is:

    v = g * t f

    where g is the acceleration of gravity (approximately 10 m/s/s on Earth; its exact value is 9.8 m/s/s). The equation above can be used to calculate the velocity of the object after a given amount of time.

    Example

    t = 6 s

    2v = (10 m/s) * (6 s) = 60 m/s f

    t = 8 s

    2v = (10 m/s) * (8 s) = 80 m/s f

How Far?

    The distance which a free-falling object has fallen from a position of rest is also dependent upon the time of fall. The distance fallen after a time of t seconds is given by the formula

    2d = 0.5 * g * t

    where g is the acceleration of gravity (approximately 10 m/s/s on Earth; its exact value is 9.8 m/s/s). This is the equation we got from the Ball Toss experiment. The equation above can be used to calculate the distance traveled by the object after a given amount of time. Example

    t = 1 s

    22d = (0.5) * (10 m/s) * (1 s) = 5 m

    t = 2 s

    22d = (0.5) * (10 m/s) * (2 s) = 20 m

    t = 5 s

    22d = (0.5) * (10 m/s) * (5 s) = 125 m

    The diagram below (not drawn to scale) shows the results of several distance calculations for a free-falling object dropped from a position of rest.

    The Big Misconception

    An earlier section of this lesson, stated that the acceleration of a free-falling object

    on Earth is 10 m/s/s. This value (known as the acceleration of gravity) is the same

    for all free-falling objects regardless of how long they have been falling, or whether they were initially dropped from rest or thrown up into the air. Yet the question is often asked "Doesn't a massive object accelerate at a greater rate than a less massive object?" This question is a reasonable inquiry that is probably based upon personal observations made of falling objects in the physical world. After all, nearly everyone has observed the difference in rate of fall of a single piece of paper (or similar object) and a textbook. The two objects clearly travel to the ground at different rates with the massive book falling

    faster.

    The answer to the question (Doesn't a massive object accelerate at a greater rate than a less massive object?) is . . . absolutely not! That is, absolutely not, if you are considering the specific type of falling motion known as free-fall. Free-fall is the motion of objects under the sole influence of gravity; free-falling objects do not encounter air resistance. Massive objects will only fall faster than less massive objects if there is an appreciable amount of air resistance present.

    The explanation of why all objects accelerate at the same rate involves the concepts of force and mass. The details will be discussed in the near future. At that time, you will learn that the acceleration of an object is directly proportional to the force acting on it and inversely proportional to its mass. Increasing force tends to increase acceleration while increasing mass tends to decrease acceleration. Thus, the greater force on massive objects is offset by the inverse influence of greater mass. So all objects, regardless of their mass, free-fall at the same rate of acceleration.

Questions - Answer in Complete Sentences!

    1. What does the term free fall mean? What force(s) are acting upon an object when it is in freefall?

    2. Will everything on earth accelerate at the exact same rate? Why or Why not?

    3. What is the numeric value for the acceleration of gravity? What is its qualitative definition (in words)? What symbol denotes it? What units are used? Be sure to answer

    all 4 parts!

4. What is an easy number to use that is “fairly accurate” for gravity we use? What is the

    correct number to use when you need to be accurate?

    5. Do we need to worry about direction when discussing gravity? Why or why not?

    6. If the velocity increases by 10 m/s each second (appears to be a constant interval), why does the ball in the picture travel further with each passing second?

    7. Describe what the velocity-time graph looks like for an object in free-fall. How can we

    find the acceleration from this?

    8. When you are talking about free fall, what values of the acceleration equation will you

    automatically know if the object starts at rest?

    9. The distance which a free-falling object has fallen is given by what formula? Then

    calculate distance for time = 10s, time = 15s, and time = 30s.

    10. Does a more massive object always fall faster than a less massive one? Explain.

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