Problem 60 - Thomas Calculus

Question

Kinetic energy The kinetic energy \(K\) of a mass is proportional to the square of its velocity \(v .\) If \(K=12,960\) joules when \(v=18 \mathrm{m} / \mathrm{sec},\) what is \(K\) when \(v=10 \mathrm{m} / \mathrm{sec} ?\)

Step-by-Step Solution

Answer

The kinetic energy when \(v = 10 \text{ m/s}\) is 4,000 joules.

Step 1: Understand the Relationship

Kinetic energy, \(K\), is given by \(K = \frac{1}{2}mv^2\), where \(m\) is mass and \(v\) is velocity. In this problem, the mass is not specified, but we know that \(K\) is proportional to \(v^2\), so we can write \(K = kv^2\) for some constant \(k\).

Step 2: Find the Constant of Proportionality

Given that \(K = 12,960\) joules when \(v = 18 \text{ m/s}\), substitute these values into the equation \(K = kv^2\):\[12,960 = k \times 18^2\]Calculate \(18^2\) to solve for \(k\):\[12,960 = k \times 324\]\[k = \frac{12,960}{324}\]\[k = 40\]

Step 3: Calculate Kinetic Energy for New Velocity

Now that we have \(k = 40\), use it to find \(K\) when \(v = 10 \text{ m/s}\). Substitute \(v = 10\) into the equation \(K = kv^2\):\[K = 40 \times 10^2\]\[K = 40 \times 100\]\[K = 4,000 \text{ joules}\]

Step 4: Verify the Calculation

Recheck the calculations step-by-step:1. \(18^2 = 324\)2. \(12,960 \div 324 = 40\) confirms our constant \(k\).3. \(10^2 = 100\), then \(40 \times 100 = 4,000\).All calculations are correct, confirming the answer.

Key Concepts

Proportionality

Proportionality is a key concept in understanding how two quantities relate to each other. In physics, when we say a quantity is proportional to another, it means there is a consistent multiplier, or ratio, linking them. For example, when we say kinetic energy is proportional to the square of velocity, it means:

• As the velocity increases, the kinetic energy increases as well.
• The square of the velocity can determine how much the kinetic energy changes.
• A constant of proportionality relates these changes consistently, allowing us to write an equation that describes this relationship.

Using proportionality helps in predicting the outcome when making changes to one of the variables, without knowing every other detail (like mass, in some problems). This concept is vital in physics, economics, biology, and many other fields, offering a simple yet powerful way to model real-world dynamics.

Velocity

Velocity is the speed of an object in a particular direction. It's a vector quantity, which means it has both magnitude (how fast something is moving) and direction (where it's going). In the kinetic energy equation, velocity plays a crucial role:

• Kinetic energy is dependent on the square of the velocity (\(v^2\)).
• If the velocity of an object changes, its kinetic energy changes dramatically, given that velocity is squared in the equation.
• A small change in velocity results in a much larger change in kinetic energy due to the square relationship.

Understanding how velocity affects kinetic energy allows students to see why a slight increase in speed can result in a significant increase in the energy of moving objects. For instance, when a car accelerates, not only does it move faster, but its kinetic energy increases substantially, impacting fuel consumption and safety measures.

Constant of Proportionality

The constant of proportionality is what links the proportional relationship between two variables mathematically. It provides the specific ratio or factor needed to go from one variable to the other. In the kinetic energy equation, the constant of proportionality is derived as follows:

• We start with the general proportionality statement: kinetic energy (\(K\)) is proportional to the square of velocity (\(v^2\)), so \(K = kv^2\).
• To find \(k\), plug in known values, like the kinetic energy and velocity, and solve the equation.
• In the provided example, from \(12,960 = k \times 18^2\), solving gives \(k = 40\). This \(k\) is then used for computations involving different velocities.

This constant allows us to compute kinetic energy for other velocities without knowing other factors, such as mass. Once it’s calculated, that constant simplifies future calculations, providing a straightforward way to link velocity and kinetic energy across various situations.