Kinematic Equations and Free Fall

A free-falling object is an object which is falling under the influence of gravity. That is, that any object which is moving and being acted upon only be the force of gravity is said to be "in a state of free fall." 

Such an object will experience a downward acceleration of 10 m/s². Whether the object is falling downward or rising upward towards its peak, if it is under the sole influence of gravity, its acceleration value is 10 m/s².

Like any moving object, the motion of an object in free fall can be described by four kinematic equations. The kinematic equations which describe any object's motion are:

The symbols in the above equation have a specific meaning: the symbol d stands for the displacement; the symbol t stands for the time; the symbol a stands for the acceleration of the object; the symbolv_i stands for the initial velocity value; and the symbol v_f stands for the final velocity - free fall.

The application of these four equations to the motion of an object in free fall can be facilitated by a proper understanding of a few conceptual characteristics of free fall motion. These concepts are described as follows:

• An object in free fall experiences an acceleration of -10 m/s² . (The - sign indicates a downward acceleration.) Whether explicitly stated or not, the value of the acceleration in the kinematic equations is -10 m/s² for any freely falling object.
• If an object is merely dropped (as opposed to being thrown) from an elevated height to the ground below, then the initial velocity of the object is 0 m/s.
• If an object is projected upwards in a perfectly vertical direction, then it will slow down as it rises upward. The instant at which it reaches the peak of its trajectory, its velocity is 0 m/s. This value can be used as one of the motion parameters in the kinematic equations; for example, the final velocity (v_f) after traveling to the peak would be assigned a value of 0 m/s.
• If an object is projected upwards in a perfectly vertical direction, then the velocity at which it is projected is equal in magnitude and opposite in sign to the velocity which it has when it returns to the same height. That is, a ball projected vertically with an upward velocity of +30 m/s will have a downward velocity of -30 m/s when it returns to the same height.

These four principles and the four kinematic equations can be combined to solve problems involving the motion of free falling objects. The two examples below illustrate application of free fall principles to kinematic problem-solving. 

Diagram:

Given: vi = 0.0 m/s d = -8.52 m a = - 10 m/s2

Find: t = ??

The next step involves identifying a kinematic equation which would allow you to determine the unknown quantity. There are four kinematic equations to choose from. In general, you will always choose the equation which contains the three known and the one unknown variable. In this specific case, the three known variables and the one unknown variable are d, v_i, a, and t. Thus, you will look for an equation which has these four variables listed in it.  Inspect the four equations above and find that the equation on the top left contains all four variables.

Once the equation is identified and written down, the next step involves substituting known values into the equation and using proper algebraic steps to solve for the unknown information. These steps are shown below.

The solution above reveals that the paint can will will fall for a time of 1.32 seconds before hitting the ground. (Note that this value is rounded to the third digit.)  

Once more, the solution to this problem begins by the construction of an informative diagram of the physical situation. This is shown below. The second step involves the identification and listing of known information in variable form. You might note that in the statement of the problem, there is only one piece of numerical information explicitly stated: 26.2 m/s. The initial velocity (v_i) of the kid is +26.2 m/s. (The + sign indicates that the initial velocity is an upwards velocity).  Note that the v_f value can be inferred to be 0 m/s since the final state of the vase is the peak of its trajectory. The acceleration (a) of the student is -10 m/s². The next step involves the listing of the unknown (or desired) information in variable form. In this case, the problem requests information about the displacement of the poor student (the height to which they rise above their starting height). So d is the unknown information. The results of the first three steps are shown in the table below.

Diagram:

Given: vi = 26.2 m/s vf = 0 m/s a = -10 m/s2

Find: d = ??

The next step involves identifying a kinematic equation which would allow you to determine the unknown quantity. There are four kinematic equations to choose from. Again, you will always search for an equation which contains the three known variables and the one unknown variable. In this specific case, the three known variables and the one unknown variable are v_i, v_f, a, and d. An inspection of the four equations above reveals that the equation on the top right contains all four variables.

Once the equation is identified and written down, the next step involves substituting known values into the equation and using proper algebraic steps to solve for the unknown information. This step is shown below.

The solution above reveals that the unfortunate child will travel upwards for a displacement of 34.32 meters before reaching its peak. (Note that this value is rounded to the third digit.)