FREE FALL: CONCEPT, EQUATIONS, SOLVED EXERCISES - PHYSICAL - 2024

The free fall is the vertical movement an object undergoes when he is dropped from a certain height near the surface of the Earth. It is one of the simplest and most immediate movements known: in a straight line and with constant acceleration.

All objects that are dropped, or that are thrown vertically up or down, move with the acceleration of 9.8 m / s ² provided by Earth's gravity, regardless of their mass.

Free fall from a cliff. Source: Pexels.com.

This fact may be accepted today without problems. However understanding the true nature of free fall took a while. The Greeks had already described and interpreted it in a very basic way by the 4th century BC.

Concept of free fall of bodies

Aristotle's ideas

Aristotle, the great philosopher of classical antiquity, was one of the first to study free fall. This thinker observed that a coin fell faster than a feather. The feather flutters as it falls, while the coin quickly makes its way to the ground. In the same way, a sheet of paper also takes its time to reach the floor.

Therefore, Aristotle had no doubts in concluding that the heaviest objects were faster: a 20 kilo rock should fall faster than a 10 gram pebble. Greek philosophers did not usually do experiments, but their conclusions were based on observation and logical reasoning.

However, this idea of ​​Aristotle, although apparently logical, was actually wrong.

Now let us do the following experiment: the sheet of paper is made into a very compact ball and simultaneously dropped from the same height as the coin. Both objects are observed to hit the ground at the same time. What could have changed?

As the paper crumpled and compacted its shape changed, but not its mass. The spread paper has more surface exposed to the air than when it is compacted into a ball. This is what makes the difference. Air resistance affects the larger object more and reduces its speed when falling.

When air resistance is not considered, all objects hit the ground at the same time as long as they are dropped from the same height. The Earth provides them with a constant acceleration of approximately 9.8 m / s ².

Galileo questioned Aristotle

Hundreds of years passed after Aristotle established his theories about motion, until someone dared to question his ideas with real experiments.

Legends say that Galileo Galilei (1564 - 1642) studied the fall of different bodies from the top of the Tower of Pisa and recognized that they all fell with the same acceleration, although he did not explain why. Isaac Newton would take care of that years later.

It is not certain that Galileo actually went up to the Tower of Pisa to do his experiments, but it is certain that he dedicated himself to doing them systematically with the help of an inclined plane.

The idea was to roll balls downhill and measure the distance traveled to the end. Afterward, I gradually increased the incline gradually, making the incline plane vertical. This is known as "gravity dilution."

Currently it is possible to verify that the pen and the coin land simultaneously when they are dropped from the same height, if the air resistance is not considered. This can be done in a vacuum chamber.

Free fall motion equations

Once convinced that the acceleration is the same for all bodies released under the action of gravity, it is time to establish the necessary equations to explain this motion.

It is important to emphasize that air resistance is not taken into account in this first movement model. However, the results of this model are very accurate and close to reality.

In everything that follows the particle model will be assumed, that is, the dimensions of the object are not taken into account, assuming that all the mass is concentrated in a single point.

For a uniformly accelerated rectilinear motion in the vertical direction, the y-axis is taken as the reference axis. The positive sense is taken up and the negative down.

Kinematic magnitudes

Thus, the equations of position, velocity, and acceleration as a function of time are:

Acceleration

Position as a function of time:

Where y _o is the initial position of the mobile and v _o is the initial velocity. Remember that in the upward vertical throw the initial velocity is necessarily different from 0.

Which can be written as:

With Δ y being the displacement made by the mobile particle. In units of the International System, both the position and the displacement are given in meters (m).

Speed ​​as a function of time:

Speed ​​as a function of displacement

It is possible to deduce an equation that links the displacement with the speed, without time intervening in it. For this, the time of the last equation is cleared:

The square is developed with the help of the notable product and terms are regrouped.

This equation is useful when you do not have time, but instead you have speeds and displacements, as you will see in the section on worked out examples.

Examples

The attentive reader will have noticed the presence of the initial velocity v _o. The previous equations are valid for vertical movements under the action of gravity, both when the object falls from a certain height, and if it is thrown vertically up or down.

When the object is dropped, simply set v _o = 0 and the equations are simplified as follows.

Acceleration

Position as a function of time:

Speed ​​as a function of time:

Speed ​​as a function of displacement

We make v = 0

Flight time is how long the object lasts in the air. If the object returns to the starting point, the rise time is equal to the descent time. Therefore, the flight time is 2. t max.

Is t _max twice the total time the object lasts in air? Yes, as long as the object starts from a point and returns to it.

If the launch is made from a certain height above the ground and the object is allowed to proceed towards it, the flight time will no longer be twice the maximum time.

Solved exercises

In solving the exercises that follow, the following will be considered:

1-The height from where the object is dropped is small compared to the radius of the Earth.

2-Air resistance is negligible.

3-The value of the acceleration of gravity is 9.8 m / s ²

4-When dealing with problems with a single mobile, preferably y _o = 0 is chosen at the starting point. This usually makes the calculations easier.

5-Unless otherwise stated, the vertical upward direction is taken as positive.

6-In the combined ascending and descending movements, the equations applied directly offer the correct results, as long as the consistency with the signs is maintained: upward positive, downward negative and gravity -9.8 m / s ² or -10 m / s ² if rounding is preferred (for convenience when calculating).

Exercise 1

A ball is thrown vertically upward with a velocity of 25.0 m / s. Answer the following questions:

a) How high does it rise?

b) How long does it take to reach your highest point?

c) How long does it take for the ball to touch the surface of the earth after it reaches its highest point?

d) What is your speed when you return to the level you started from?

Solution

c) In the case of a level launch: t _flight = 2. t _max = 2 x6 s = 5.1 s

d) When it returns to the starting point, the velocity has the same magnitude as the initial velocity but in the opposite direction, therefore it must be - 25 m / s. It is easily checked by substituting values ​​into the equation for velocity:

Exercise 2

A small mail bag is released from a helicopter that is descending with a constant speed of 1.50 m / s. After 2.00 s calculate:

a) What is the speed of the suitcase?

b) How far is the suitcase under the helicopter?

c) What are your answers for parts a) and b) if the helicopter is rising with a constant speed of 1.50 m / s?

Solution

Paragraph a

When leaving the helicopter, the bag carries the initial velocity of the helicopter, therefore v _o = -1.50 m / s. With the indicated time, the speed has increased thanks to the acceleration of gravity:

Section b

Let's see how much the suitcase has dropped from the starting point in that time:

Y _o = 0 has been selected at the starting point, as indicated at the beginning of the section. The negative sign indicates that the suitcase has descended 22.6 m below the starting point.

Meanwhile the helicopter has descended with a speed of -1.50 m / s, we assume with constant speed, therefore in the indicated time of 2 seconds, the helicopter has traveled:

Therefore after 2 seconds, the suitcase and the helicopter are separated by a distance of:

Distance is always positive. To highlight this fact, the absolute value is used.

Section c

When the helicopter rises, it has a velocity of + 1.5 m / s. With that speed the suitcase comes out, so that after 2 s it already has:

The speed turns out to be negative, since after 2 seconds the suitcase is moving downwards. It has increased thanks to gravity, but not as much as in section a.

Now let's find out how much the bag has descended from the starting point during the first 2 seconds of travel:

Meanwhile, the helicopter has risen from the starting point, and has done so with constant speed:

After 2 seconds the suitcase and the helicopter are separated by a distance of:

The distance that separates them is the same in both cases. The suitcase travels less vertical distance in the second case, because its initial velocity was directed upwards.

References

1. Kirkpatrick, L. 2007. Physics: A Look at the World. 6 ^ta Editing abbreviated. Cengage Learning. 23 - 27.
2. Rex, A. 2011. Fundamentals of Physics. Pearson. 33 - 36
3. Sears, Zemansky. 2016. University Physics with Modern Physics. 14 ^th. Ed. Volume1. 50 - 53.
4. Serway, R., Vulle, C. 2011. Fundamentals of Physics. 9 ^na Ed. Cengage Learning. 43 - 55.
5. Wilson, J. 2011. Physics 10. Pearson Education. 133-149.