Free fall motion
In Physics, motion of a free falling object is a classic, but still an elementary example of motion along a straight line with constant acceleration. It is a motion under influence of gravity alone. The gravity is one of the most interesting phenomena studied by Physics.
We exclude influences of any other factors especially the effect of the air in which this motion usually occurs.
In these days of space travels it is worth mentioning that we will study the free fall of an object on to Earth (not a different planet) and not too far from its surface. The free fall in any other place (like the Moon) is governed by the same physics laws, only numerical values of acceleration due to gravity are different.
Free fall motion is the motion in a vertical direction as seen by an observer on the Earth’s surface. It is motion with constant acceleration g = 9.81 m/s2 directed downwards. If we drop a stone from the window of a 4 story building (do not perform such experiments unless the ground around the building is not accessible to anyone), that is from a height of about 12 m, you may be interested to find that:
t - time needed to reach the level of the ground,
vf – the final velocity it will have at the moment of reaching the ground.
As this is motion with constant acceleration we can use Eq. M1.30
with initial displacement x0 = 0 and initial velocity v0 = 0. The acceleration a in this equation we denote a g, which should remind us that it is gravity’s acceleration, and displacement x we replace with letter h, which is customarily used to denote the height. So the Eq. M1.30 will be transposed into
The free fall is depicted schematically on Fig. M1.4. Notice, that the downward direction of motion is chosen to be positive. It implies that motion downwards is the natural direction of the free falling object.
Fig.M1.4
From Eq. M1.32 we can find the time needed to fall a distance of h.
We use here notation x1/2 for the square root of x. Generally
This “power” notation is often more handy in writing and should not be confusing to anyone from the high school level up.
The final velocity of a falling object can be calculated by substituting time from Eq. M1.33 into Eq. M1.25
or, after rearranging
and for those who prefer notation with the symbol of a square root,
(M1.34)Physics and its equations describing the free fall motion can used to determine the height of a building or how deep a well is.
We have to measure the time for which the object is in motion during the free fall and knowing it we can calculate, using the Eq. M1.33, the distance traveled, that is the height of the building or the depth of the well.
How to measure this depth of the well if we do not see thr bottom, so we cannot directly measure the time of the free fall of an arbitrary object, usually a stone?
The knowledge of Physics laws will help us. The procedure of the experiment allowing such measurements is depicted in Fig. M1.5.
Fig. M1.5
The stone is dropped and the stopwatch is started. We wait till the sound of a stone dropping into water is heard. The time from the moment of dropping the stone to the moment of hearing the sound is the sum of free falling time tf and time ts, that the sound needed to travel from the bottom to the surface.
tT=tf+ts
Denoting the distance from the top to the bottom of the well by h, we can write three equations
g – gravity acceleration,
vs – speed of sound,
This is a classical and elementary situation when solving problems in Physics – an algebraic set of three equations with three unknowns. Solving them should not pose any difficulties, so we stop at this point. There will be few another examples of free fall between problems at the end of this chapter.