Problem 50 - Elementary Algebra

Question

Applications involving variation. The distance, \(d\), an object in free fall drops varies directly with the square of the time, \(t\), that it has been falling. If an object in free fall drops 36 feet in 1.5 seconds, then how far will it have fallen in 3 seconds? Hooke's law suggests that the extension of a hanging spring is directly proportional to the weight attached to it. The constant of variation is called the spring constant.

Step-by-Step Solution

Answer

The object falls 144 feet in 3 seconds.

Step 1: Understand the Variation Formula

Since the distance covered in free fall varies directly with the square of the time, we write this relationship as a direct variation formula: \(d = kt^2\), where \(k\) is the constant of variation.

Step 2: Use Given Data to Find the Constant

We are given that the distance \(d = 36\) feet when the time \(t = 1.5\) seconds. Substitute these values into the formula to find \(k\):\[36 = k(1.5)^2\]Which simplifies to:\[36 = k imes 2.25\] Thus, \(k = \frac{36}{2.25} = 16\).

Step 3: Calculate the New Distance

We need to find the distance \(d\) when time \(t = 3\) seconds, using the constant \(k = 16\) from Step 2. Substitute these into the formula:\[d = 16(3)^2\]This simplifies to:\[d = 16 imes 9 = 144\]So, the object will have fallen 144 feet in 3 seconds.

Key Concepts

Free Fall

Free fall refers to the motion of an object falling solely under the influence of gravity, with no other forces acting upon it, like air resistance. When an object is in free fall, it accelerates uniformly due to gravity. This means its velocity increases by a constant amount every second.
The distance that the object falls during free fall can change with time. Importantly, it varies directly with the square of the time it has been falling. This direct variation is expressed in the equation:

• \( d = kt^2 \)

where:

• \(d\) is the distance fallen
• \(t\) is the time of fall
• \(k\) is the constant of variation

In simple terms, if you double the time, the distance fallen increases fourfold, because distance grows with the square of time. This principle is very useful in predicting how far objects fall over different time periods when only gravity is acting.

Hooke's Law

Hooke's law is a principle that applies to elastic materials like springs. It states that the elongation, or stretch, of the spring is directly proportional to the weight or force applied to it. This means if the force increases, the stretch increases in a linear fashion.
The mathematical expression of Hooke's law is:

• \( F = kx \)

where:

• \(F\) is the force applied to the spring
• \(x\) is the extension or compression of the spring
• \(k\) is the spring constant, representing the stiffness of the spring

Understanding Hooke's law helps us estimate how much a spring will stretch under a certain force. Each spring has a unique spring constant, which must be experimentally determined to use the formula accurately.

Constant of Variation

In both direct and inverse variation, the constant of variation is a crucial part of the equation. It signifies how one quantity changes in relation to another. In direct variation, as seen in the free fall example, the relationship is expressed as:

• \( y = kx \)

Here:

• \(k\) is the constant of variation

This constant tells us the ratio between two directly varying quantities. In simpler terms, it explains how one variable changes with respect to another. For instance, when talking about free fall, the constant offers insights into how far an object travels in a given time span. If you have two sets of data in a direct variation, finding the constant helps in predicting the value of one quantity given the other.

Distance-Time Relationship

The distance-time relationship is an essential aspect of understanding motion. It describes how distance traveled by an object changes over time. For objects moving at constant speeds, this relationship is linear, but when acceleration is involved, as in the case of free fall, the relationship is quadratic.
The equation we use to describe the distance-time relationship in free fall is:

• \(d = kt^2\)

This quadratic relationship means that the distance changed is proportional to the square of the time interval. In practical terms, if you consider an object falling under gravity, the longer it falls, the faster it goes, resulting in a greater distance covered as time goes on.
Recognizing this relationship aids in predicting how far and how fast an object will move over time, which is useful in both physics studies and everyday problem-solving scenarios involving motion.