Problem 8 - Chemistry

Question

A baseball with a mass of \(450 \mathrm{~g}\) has a kinetic energy of \(406 \mathrm{~J} .\) Calculate the velocity of the baseball in units of \(\mathrm{m} / \mathrm{s}\).

Step-by-Step Solution

Answer

The velocity of the baseball is approximately 42.48 m/s.

Step 1: Convert Mass to Kilograms

The mass of the baseball is given in grams. We need to convert this mass to kilograms for use in the formula. This is done by dividing by 1000:\[ m = \frac{450}{1000} = 0.45 \text{ kg} \]

Step 2: Understand Kinetic Energy Formula

The formula for kinetic energy (KE) is \( KE = \frac{1}{2}mv^2 \), where \( m \) is mass and \( v \) is velocity. We can use this formula to find \( v \).

Step 3: Rearrange the Formula to Solve for Velocity

We need to solve for \( v \). Begin by rearranging the kinetic energy formula:\[ KE = \frac{1}{2} m v^2 \]Multiply both sides by 2 to get rid of the fraction:\[ 2 \cdot KE = m v^2 \]Now, divide both sides by \( m \):\[ v^2 = \frac{2 \cdot KE}{m} \]Take the square root of both sides to solve for \( v \):\[ v = \sqrt{\frac{2 \cdot KE}{m}} \]

Step 4: Substitute Values into the Formula

Substitute the given kinetic energy (406 J) and the converted mass (0.45 kg) into the rearranged formula:\[ v = \sqrt{\frac{2 \cdot 406}{0.45}} \]

Step 5: Calculate the Velocity

Calculate the velocity:\[ v = \sqrt{\frac{812}{0.45}} \]\[ v = \sqrt{1804.44} \]\[ v \approx 42.48 \text{ m/s} \]

Key Concepts

Mass Conversion

When dealing with scientific equations, especially in physics, it's crucial to use consistent units. In the original exercise, we are dealing with kinetic energy, which is typically calculated using the SI system of units. Here, the mass of the baseball is originally provided in grams (450 g). To align with the standard units of the kinetic energy formula, akin to the system typically used in scientific calculations, masses should be in kilograms.

To convert grams to kilograms, you need to remember this simple conversion:

• 1 kilogram = 1000 grams

The calculation is straightforward: divide the number of grams by 1000. By performing this division, we convert 450 g into:\[ 0.45 ext{ kg} \]This allows us to use the mass directly in the subsequent kinetic energy calculations without any unit errors. This step is vital in ensuring all quantities used are compatible within the standard formula, helping avoid potential errors during computations.

Velocity Calculation

Velocity is a measure of the speed and direction of an object's movement. In the context of kinetic energy, which measures how much energy an object has due to its motion, velocity plays a key role.

In the solved exercise, you're tasked with determining the velocity of a baseball, given its kinetic energy and mass. Once the mass conversion is completed, the next step involves manipulating the kinetic energy formula to find the velocity. The kinetic energy formula:\[ KE = \frac{1}{2}mv^2 \]deals with velocity as part of its equation. The task is to rearrange this formula to isolate the velocity.

You begin by eliminating the fraction:

• Multiply both sides by 2 to simplify: \[ 2 \cdot KE = mv^2 \]
• Then, divide both sides by the mass (\(m\)): \[ v^2 = \frac{2 \cdot KE}{m} \]
• The final step involves taking the square root to solve for \(v\): \[ v = \sqrt{\frac{2 \cdot KE}{m}} \]

This rearrangement allows direct calculation of velocity once the kinetic energy and mass values are substituted.

Kinetic Energy Formula

The kinetic energy formula is a fundamental component in physics and an essential tool for understanding the movement of objects. In this formula, kinetic energy (KE) is expressed as a function of an object's mass and velocity:\[ KE = \frac{1}{2}mv^2 \]Here:

• \( KE \) is the kinetic energy in joules.
• \( m \) represents the mass in kilograms.
• \( v \) is the velocity in meters per second.

The formula illustrates how kinetic energy depends on both mass and velocity. Notice that velocity is squared, indicating its strong influence on the kinetic energy. This means that small changes in speed can lead to large changes in the energy.

This exercise guides you through rearranging and applying this formula to find the velocity when the kinetic energy and mass are known. By doing so, you grasp not just how to plug in numbers but also the conceptual relationship between mass, velocity, and energy the formula captures. Understanding this connection is key to mastering kinetic energy calculations in physics.