Problem 60 - College Algebra

Question

For the following exercises, use the given information to answer the questions. The kinetic energy \(K\) of a moving object varies jointly with its mass \(m\) and the square of its velocity \(v\) . If an object weighing 40 kilograms with a velocity of 15 meters per second has a kinetic energy of 1000 joules, find the kinetic energy if the velocity is increased to 20 meters per second.

Step-by-Step Solution

Answer

The kinetic energy is 1777.78 joules.

Step 1: Understand the Relationship

Kinetic energy (\(K\)) varies jointly with the mass (\(m\)) and the square of the velocity (\(v\)). This relationship can be expressed as \(K = k \, m \, v^2\) where \(k\) is a constant of proportionality.

Step 2: Substitute Known Values to Find the Constant

Using the information given, substitute \(m = 40\), \(v = 15\), and \(K = 1000\) into the equation: \[1000 = k \, (40) \, (15)^2\]Solve for \(k\).

Step 3: Calculate the Constant of Proportionality

Substitute to find \(k\): \[1000 = k \, (40) \, (225)\]\[1000 = 9000k\]\[k = \frac{1000}{9000} = \frac{1}{9}\]

Step 4: Apply the Formula with New Velocity

Use the formula \(K = k \, m \, v^2\) with the updated velocity \(v = 20\). Using \(k = \frac{1}{9}\) and \(m = 40\): \[K = \frac{1}{9} \, (40) \, (20)^2\]

Step 5: Calculate the New Kinetic Energy

Substitute and solve: \[K = \frac{1}{9} \, (40) \, (400)\]\[K = \frac{1}{9} \, (16000)\]\[K = 1777.78 \text{ Joules}\]

Key Concepts

Understanding Joint Variation

Joint variation refers to a situation where one quantity varies directly as the product of two or more other quantities. In the case of kinetic energy, the concept of joint variation helps us understand how kinetic energy (\(K\)) depends on both the mass (\(m\)) and the square of its velocity (\(v^2\)).
The term "jointly" signifies that both mass and velocity factor into determining kinetic energy. This means if either mass or velocity changes, the kinetic energy will change accordingly.
Thus, the relationship can be mathematically expressed as:

• \(K = k \, m \, v^2\)

where \(k\) is a constant. This equation neatly encapsulates the joint dependence of kinetic energy on both mass and velocity squared.

Exploring the Proportionality Constant

The proportionality constant \(k\) represents the rate at which kinetic energy varies with respect to mass and the square of velocity. In the equation \(K = k \, m \, v^2\), \(k\) is crucial in helping us compute the actual kinetic energy from given values.
Let's dive into the calculation of this constant. In our example, with known values \(m = 40\), \(v = 15\), and \(K = 1000\), we substitute them to find \(k\):

• \[1000 = k \, (40) \, (15)^2\]
• \[1000 = k \, 9000\]
• \[k = \frac{1000}{9000} = \frac{1}{9}\]

This constant \(\frac{1}{9}\) helps maintain consistency in the relationship, making future calculations for kinetic energy, based on different velocities, straightforward and accurate.

Understanding the Square of Velocity

The kinetic energy formula involves the square of velocity, \(v^2\), which means that any change in velocity significantly affects the kinetic energy. Squaring the velocity emphasizes how much more energy an object can have with even small increases in speed.
This aspect of the formula helps us see why vehicles at high speeds can possess much more kinetic energy, underscoring the importance of speed control for safety. Mathematically, it tells us that doubling the velocity leads to a fourfold increase in kinetic energy, illustrating the powerful impact of velocity on kinetic energy calculations.

Interplay of Mass and Velocity

Mass and velocity are the two critical components influencing kinetic energy. While the mass represents the amount of matter in the object, velocity indicates how fast the object is moving.
In our kinetic energy equation \(K = k \, m \, v^2\):

• The mass \(m\) linearly affects energy. Double the mass will double the kinetic energy, given a constant velocity.
• Velocity’s effect, however, is exponential because we use the square of velocity \(v^2\), making it a dominant factor in energy levels.

This interplay shows why both heavy and fast-moving objects can have significant kinetic energy, each aspect contributing uniquely to the total energy.