Free Falling Object

Free Falling

Below we define and discuss free falling and its characteristics.

When objects are in free fall, these objects are assumed to fall within a vacuum. As a result, this motion is defined by two characteristics:

1. Objects do not experience air resistance.
2. Objects accelerate towards earth at \( 9.8\,\mathrm{\frac{m}{s^2}}. \)

Weight of a Free-Falling Object

When an object is in free fall, the gravitational force acting on it is said to be equal to the weight of the object. The formula for weight is given by the equation $$\begin{align}W&=mg\\W&=F\\\end{align},$$ where \(m\) is mass, \(g\) is the acceleration due to gravity, and \(F) is the force of weight.

Equations of a Free-Falling Object

Variables associated with the motion of free falling can be calculated through the use of kinematics.

In the field of kinematics, there are four corresponding variables of motion. These variables are

1. Displacement
2. Velocity
3. Acceleration
4. Time

Thus, let us define each variable starting with displacement.

The mathematical formula corresponding to this definition is $$\Delta{y}=\Delta{v}\,\Delta{t}$$ where \( \Delta{v} \) is velocity and \( t \) is time. Displacement has a SI unit of \(\mathrm{m}\).

The mathematical formula corresponding to this definition is

$$v=\frac{\Delta{y}}{\Delta{t}}$$ where \( \Delta{y} \) is displacement and \( t \) is time. Velocity has a SI unit of \( \mathrm{\frac{m}{s}} \).

The mathematical formula corresponding to this definition is

$$a=\frac{\Delta{v}}{\Delta{t}}$$ where \( v \) is velocity and \( t \) is time. Acceleration has a SI unit of \( \mathrm{\frac{m}{s^2}} \).

Time does not change relative to an object's type of motion and is measured in \( s. \)

Relationship between Variables

Consequently, three equations describe the relationship between these variables. These equations are the kinematic equations of motion.

The velocity equation,

$$v=v_{o} + g{t}.$$

The displacement equation,

$$\Delta{y} =v_o{t}+\frac{1}{2}{g}t^2.$$

The velocity squared equation,

$$v^2=v_{o}^2 +2g\Delta{y}.$$

The average velocity equation,

$$v_{avg}=\frac{v+v_o}{2}.$$

where \( v \) is final velocity, \( v_o \) is the initial velocity, \( g \) is acceleration due to gravity, \( t \) is time, and \( \Delta{y} \) is displacement.

Note that the gravitational constant, \( g \), has replaced acceleration, \( a \), in these equations and \( \Delta{x} \) has been replaced by \( \Delta{y} \) because we are working along the y-axis.

Acceleration of a Free-Falling Object

The acceleration of an object in free fall is equal to gravity. This can be seen by rearranging Newton's second law of motion, \( F=ma \), and solving for acceleration.

$$\begin{align}F&=ma\\a&=\frac{F}{m}=\frac{W}{m}=\frac{mg}{m}=g\\a&=g.\\\end{align}$$

This implies that objects will accelerate at the same rate regardless of their shape, size, or mass. However, we must understand that this neglects air resistance. For example, if air resistance was not neglected, a ball and sheet of paper in free fall would not accelerate at the same rate. The ball would accelerate faster and reach the ground first because it has a larger mass and larger momentum in comparison to the sheet of paper.

Speed and Velocity of a Free-Falling Object

Objects in free-fall accelerate downward toward the Earth's surface. This downward motion indicates negative acceleration. As acceleration tells us how fast velocity changes every second, the downward direction also implies that velocity will be negative. This can be explained by the fact that acceleration and velocity are vector quantities, with magnitude and direction. As the direction is downward, a negative value is implied. However, although velocity is negative, speed will be positive. As speed is a scalar quantity, with only magnitude, it is not affected by direction. Therefore, for an object in free fall, its velocity becomes more negative every second, while its speed becomes more positive every second.

To calculate the velocity and speed of an object in free fall, the kinematic equations, mentioned earlier, can be used.

Position Function for Free Falling Objects

To calculate an object's position, during free fall, with respect to time, the equation \( \Delta{y} =v_o{t}+\frac{1}{2}{g}t^2 \) can be used. However, we can note that objects in free fall start from rest, indicating the object's initial velocity is zero. As a result, the equation can be simplified to \( \Delta{y}=\frac{1}{2}gt^2. \) The equation \( v^2=v_{o}^2 +2g\Delta{y} \) also can be used to determine position.

Free Falling Object in Equilibrium

When an object is initially released from rest, the only force acting on it is its weight which is given by \( W=mg. \) As the object accelerates downward, it begins to experience the effects of air resistance. Air resistance opposes the force of weight, but the object's weight will be greater allowing the object to continue accelerating. However, at a certain point, the object will reach its terminal velocity, its maximum speed, where air resistance becomes equal to weight. As a result, the object is then considered to be in equilibrium because it is no longer accelerating. This can be seen in the free-body diagram below.

Free body diagram of an object falling at terminal velocity

Let us use the free-body diagram below as a visual representation of our discussion of free fall and terminal velocity.

Figure 2: A free-body diagram demonstrating an object in free fall.

Examples of how to solve objects in free fall problems

To solve free fall problems, kinematic equations can be applied to calculate an object's acceleration, velocity, speed, and position. As we have discussed free-falling objects, their motion, and the forces affecting them, let us work through some examples to gain a better understanding of each concept. Note that before solving a problem, we must always remember these simple steps:

1. Read the problem and identify all variables given within the problem.
2. Determine what the problem is asking and what formulas are needed.
3. Apply the necessary formulas and solve the problem.
4. Draw a picture if necessary to provide a visual aid.

Examples

Let us apply our new knowledge of kinematics and free fall motion to the following two examples.

Now, after solving for displacement, let's solve for a different variable in the example below.