What Are Gravitational Forces Like?: Whenever an object is dropped from a distance very near the surface of the Earth, it speeds up in a downward direction with an acceleration of magnitude 9.8...

What Are Gravitational Forces Like?: Whenever an object is dropped from a distance very near the surface of the Earth, it speeds up in a downward direction with an acceleration of magnitude 9.8 m/s2

 Two students have come up with different explanations for this phenomenon:

Kasi argues: “An object tends to move with a velocity that is proportional to the force on it and when an object falls there must be a gravitational force on it. This gravitational force in the downward direction gets larger and larger as the object falls and gets closer to the Earth. This causes the velocity to get larger and larger as the object falls, so the object undergoes a constant acceleration because the force gets larger at a constant rate as the object falls.

Heather argues: “An object tends to move with an acceleration that is proportional to the force on it. Since all falling objects accelerate at a constant rate, there must be a special gravitational force attracting it toward the center of the Earth that is essentially constant near the surface of the
Earth.

Do you agree with Kasi or with Heather? Which argument is invalid? What observations have you made that would support your opinion? 

Asked on by Ruya34

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valentin68 | College Teacher | (Level 3) Associate Educator

Posted on

Since  the acceleration is always `a =F/m` and the dependence of speed on constant accelerations is of the type `v=a*t =(F/m)*t` it means that when saying "an object is moving with a speed proportional to the force on it" both persons are right.

However the constant of proportionality between speed and force is time, which basically means for an uniform increasing speed, a constant applied force.

This means that the arguments of Heather are valid and the arguments of Kasi are false. In reality, both students are right, the force is essentially constant but at the same time is increasing when the object is approaching Earth ("essentially" is the main word here).

But at this point the reasoning of Kasi begins to fail. First the velocity does not get larger because the force becomes much larger, but because time passes (as shown above the constant of proportionality between speed and force is time). Second, since the acceleration is proportional to the applied force (`a =F/m`) it is false to say that the acceleration is constant and the force gets larger at a constant rate. (Otherwise said, the time is not involved in the definition of acceleration, therefore to relate the rate of force change to the acceleration is a nonsense).

The acceleration is constant basically because the gravitational force expression is

`F = G*(m*M)/R^2 =m*g`

where `R =6371 km =6371000 m` and a difference in the value of `R` of 10 meters or so does not change significantly the value force.

Thus the argument that is invalid here is that the acceleration is constant due to the constant rate of change in the force.

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