Problem 31 - General Chemistry

Question

A small car is traveling at twice the speed of a larger car, which has twice the mass of the smaller car. Which car has the greater kinetic energy? (Or do they both have the same kinetic energy?)

Step-by-Step Solution

Answer

The small car has the greater kinetic energy.

Step 1: Understanding Kinetic Energy Formula

Kinetic energy (KE) of an object is given by the formula: \[ KE = \frac{1}{2}mv^2 \]where \( m \) is the mass and \( v \) is the velocity of the object. Our goal is to compare the kinetic energies of two cars.

Step 2: Defining Variables

Let the mass of the smaller car be \( m \) and its speed be \( v \). Thus, the mass of the larger car will be \( 2m \) and its speed will be \( \frac{v}{2} \).

Step 3: Calculating Kinetic Energy of Smaller Car

Using the formula \( KE = \frac{1}{2}mv^2 \), we substitute \( m \) for the mass and \( v \) for the speed:\[ KE_{small} = \frac{1}{2}m(v)^2 = \frac{1}{2}mv^2 \]

Step 4: Calculating Kinetic Energy of Larger Car

For the larger car, substitute \( 2m \) for the mass and \( \frac{v}{2} \) for the speed:\[ KE_{large} = \frac{1}{2}(2m)\left(\frac{v}{2}\right)^2 \]

Step 5: Simplifying Kinetic Energy of Larger Car

Continuing from the expression for the larger car:\[ KE_{large} = \frac{1}{2} \times 2m \times \frac{v^2}{4} = \frac{1}{2} \times \frac{2m}{4} \times v^2 = \frac{1}{4}mv^2 \]

Step 6: Comparison of Kinetic Energies

Now, compare \( KE_{small} = \frac{1}{2}mv^2 \) with \( KE_{large} = \frac{1}{4}mv^2 \). Clearly, \( \frac{1}{2}mv^2 > \frac{1}{4}mv^2 \). Thus, the smaller car has the greater kinetic energy.

Key Concepts

Kinetic Energy Formula

In physics, understanding the Kinetic Energy Formula is crucial. Kinetic energy measures the energy an object possesses due to its motion. The formula is represented as \( KE = \frac{1}{2}mv^2 \), where \( KE \) stands for kinetic energy, \( m \) is the mass of the object, and \( v \) is its velocity. This formula shows that kinetic energy is directly proportional to the mass of the object and the square of its velocity.

• If the velocity doubles, the kinetic energy quadruples, because velocity is squared.
• If the mass doubles, the kinetic energy doubles.

Understanding this relationship is key to solving problems related to moving objects and comparing their energies.

Physics Problems

Physics problems often require the application of formulas to find unknown quantities or to compare different scenarios. In the case of kinetic energy, problems typically involve identifying the mass and velocity of objects. Then, using the Kinetic Energy Formula to compute their energies.
Examples might include:

• Comparing the kinetic energies of two vehicles due to changes in mass and velocity.
• Solving for one of the variables (mass or velocity) when the kinetic energy is given.

These kinds of problems help students practice not only their mathematical skills but also their conceptual understanding of how energy works in real-world scenarios.

Mass and Velocity in Kinetic Energy

When dealing with kinetic energy, two critical factors are mass and velocity. Mass refers to how much matter is in an object, while velocity refers to the speed and direction that object is traveling. In terms of kinetic energy, even small changes in velocity have a significant impact because velocity is squared in the formula. This means:

• A small increase in velocity results in a much larger increase in kinetic energy compared to the same percentage change in mass.
• For two objects with the same velocity, the one with larger mass will have more kinetic energy.

Understanding these effects helps in predicting how different variables will alter an object's kinetic energy.

Comparing Kinetic Energies

Comparing kinetic energies between two objects requires careful consideration of their masses and velocities. For instance, in the problem involving two cars, one smaller car travels faster (twice the speed), while the other is heavier (double the mass). By applying the Kinetic Energy Formula, we can determine:

• The smaller car, despite its lesser mass, has a greater kinetic energy because its speed has a much larger influence.
• Conversely, the larger car's slower speed significantly reduces its kinetic energy, more than its increased mass compensates for.

Problems like these demonstrate the complex interplay between mass and velocity in determining kinetic energy, showing how kinetic energy isn't just about size or speed alone, but the combination of both.