Problem 64 - Beginning and Intermediate Algebra

Question

Does an object fall the same distance during each second? Why or why not?

Step-by-Step Solution

Answer

No, an object falls greater distances each second due to gravitational acceleration.

Step 1: Define the Motion Context

When an object is in free fall, it is only under the influence of gravitational acceleration, which is approximately \(9.8 \, \text{m/s}^2\) near the Earth's surface.

Step 2: Initial Velocity Consideration

For an object that starts from rest, initial velocity \(v_0 = 0\, \text{m/s}\). At each subsequent second, the velocity increases due to acceleration.

Step 3: Distance Calculation for Each Second

In the first second, the object travels a distance calculated by the formula \(s = \frac{1}{2}gt^2\). By substituting for the first second, \(s_1 = \frac{1}{2} \times 9.8 \times 1^2 = 4.9\, \text{m}\).

Step 4: Distance Calculation for the Next Second

In the second second, calculate the total distance and subtract the distance covered in the first second. Thus, the distance for the second second is \(s_2 = \frac{1}{2} \times 9.8 \times 2^2 - 4.9 = 14.7 - 4.9 = 9.8\, \text{m}\).

Step 5: Compare Distances

Clearly, \(s_2 > s_1\) as \(9.8\, \text{m} > 4.9\, \text{m}\). This demonstrates that the distance traveled by the object in free fall increases each second due to acceleration by gravity.

Key Concepts

Gravitational Acceleration

In the context of free fall, gravitational acceleration plays a crucial role. It is the acceleration of an object caused by the gravitational force exerted by a massive body like the Earth. Near the Earth's surface, this acceleration is approximately constant at about \(9.8 \, \text{m/s}^2\).
This means that for every second an object is in free fall, its velocity increases by \(9.8 \, \text{m/s}\).
Gravitational acceleration is a vector quantity, pointing towards the center of the Earth. Key aspects of gravitational acceleration include:

• The value remains constant, irrespective of the object's mass.
• It pulls objects towards the Earth uniformly.
• All objects, irrespective of their mass, experience the same acceleration when only gravity acts upon them (ignoring air resistance).

Understanding gravitational acceleration is fundamental to calculating how an object's motion changes over time when in free fall.

Distance Calculation

To predict how far an object in free fall will travel, we use the formula \(s = \frac{1}{2}gt^2\).
This formula helps determine how distance relates to time under constant acceleration.
Here's how the calculation works:

• \(s\) represents distance traveled.
• \(g\) is gravitational acceleration, approximately \(9.8 \, \text{m/s}^2\).
• \(t\) is the time in seconds that the object has been falling.

In the first second, an object falls \(4.9\) meters. This is derived from \(s_1 = \frac{1}{2} \times 9.8 \times 1^2\).
For the second second, the cumulative distance is calculated using \(t = 2\), resulting in \(14.7\) meters. The object travels an additional \(9.8\) meters in the second second, as we subtract the initial \(4.9\) meters.
This calculation shows that as more time passes, the distance covered per second increases due to constant acceleration.

Velocity

The concept of velocity in free fall describes how quickly an object moves in a specified direction. With gravitational acceleration, an object starting from rest (where initial velocity \(v_0 = 0\)) continuously increases its velocity as long as it is falling.
Understanding velocity in free fall involves recognizing:

• The continuous increase in speed due to \(9.8 \, \text{m/s}^2\) acceleration.
• Velocity after one second is \(9.8 \, \text{m/s}\); after two seconds, it reaches \(19.6 \, \text{m/s}\).
• Expressing velocity as \(v = gt\), shows how linear velocity growth relates to time.

Because of this linear relationship, while acceleration remains constant, velocity allows us to predict how rapidly distance increases during free fall. The behaviors of velocity and distance are interlinked, as being discussed in free-fall scenarios.