Problem 30 - Algebra Form and Function

Question

If a ball is dropped from a high window, the distance, \(D,\) in feet, it falls is proportional to the square of the time, \(t,\) in seconds, since it was dropped and is given by $$ D=16 t^{2} $$ How far has the ball fallen after three seconds and after five seconds? Which distance is larger? Explain your answer in algebraic terms.

Step-by-Step Solution

Answer

Answer: The distance fallen after 5 seconds (400 feet) is larger than the distance fallen after 3 seconds (144 feet).

Step 1: Find distance fallen after 3 seconds

Substitute \(t=3\) into the equation \(D=16t^2\): $$ D = 16(3)^2 $$ Now, calculate the distance: $$ D = 16(9) = 144 $$ So, after 3 seconds, the ball has fallen 144 feet.

Step 2: Find distance fallen after 5 seconds

Substitute \(t=5\) into the equation \(D=16t^2\): $$ D = 16(5)^2 $$ Now, calculate the distance: $$ D = 16(25) = 400 $$ So, after 5 seconds, the ball has fallen 400 feet.

Step 3: Compare the distances

We found that the ball has fallen 144 feet after 3 seconds and 400 feet after 5 seconds. To determine which distance is larger, we can compare the two values in algebraic terms: $$ 144 < 400 $$ Thus, the distance fallen after 5 seconds (400 feet) is larger than the distance fallen after 3 seconds (144 feet).

Key Concepts

Proportional Relationships

In mathematics, a proportional relationship refers to a scenario where one quantity is related to another through a constant multiplier. This concept is evident in the relationship between the distance a falling ball travels and the time elapsed, as described in the exercise. Here, distance \(D\) is directly proportional to the square of the time \(t^2\). This means that if you were to double the time, the distance would quadruple since the time is squared in this relationship.

• The equation for this proportional relationship is given by \(D = 16t^2\).
• The constant of proportionality here is \(16\).

This means for every second the ball is in free fall, its fallen distance increases by a multiple of the square of the time elapsed, multiplied by \(16\). This reflects a real-world physical principle of gravity acting on freely falling objects.

Distance-Time Relationship

The distance-time relationship examines how far an object travels over a period of time. In this context, we are looking at a special situation where the ball falls freely under the effect of gravity. Here, the distance is not just increasing linearly with time but with the square of time due to the constant acceleration of gravity. Some key points to keep in mind:

• The formula \(D = 16t^2\) illustrates this relationship, showing that distance is proportional to the square of time.
• This quadratic relationship implies that, as time passes, the rate at which distance increases grows larger.

For example, between 3 and 5 seconds, the ball doesn't just fall twice as far; it falls more because the increment depends on the square of the time difference. This characteristic is typical of objects in free fall, where acceleration due to gravity continuously increases its velocity.

Mathematical Modeling

Mathematical modeling involves creating equations and formulas to represent real-world phenomena. The equation \(D = 16t^2\) is a prime example of a mathematical model. It succinctly captures the effect of gravity on a falling object:

• By using this mathematical equation, we can predict the distance a ball will fall over any given time \(t\).
• Mathematical models offer a way to analyze data and test scientific hypotheses by simulating real-world scenarios.

Understanding this model allows us to predict outcomes without physically performing each experiment. For instance, by substituting different time values into the equation, you could calculate how far the ball would fall at any moment within the given constraints. This makes mathematical modeling a powerful tool in science and engineering, providing insights into systems and phenomena efficiently.