Question

How fast (in meters per second) must an iron ball with a mass of \(56.6 \mathrm{~g}\) be traveling to have a kinetic energy of \(15.75 \mathrm{~J} ?\)

Short answer

The ball must travel at approximately 23.59 m/s.

Step-by-step solution

Understand the equation for kinetic energy

The kinetic energy (KE) of an object is given by the equation \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the object in kilograms and \( v \) is the speed in meters per second. In this problem, \( KE \) is given as 15.75 J and mass \( m \) is 56.6 g, which needs to be converted to kilograms.

Convert mass from grams to kilograms

Convert the mass of the iron ball from grams to kilograms by dividing by 1000. \[ m = \frac{56.6}{1000} = 0.0566 \text{ kg} \]

Rearrange the equation to solve for speed

The equation for kinetic energy is \( KE = \frac{1}{2}mv^2 \). Rearrange this equation to solve for \( v \):\[ v = \sqrt{\frac{2KE}{m}} \]

Substitute the known values into the equation

Now substitute the known values into the equation:\[ v = \sqrt{\frac{2 \times 15.75}{0.0566}} \]

Calculate the speed

Calculate the speed by following through the arithmetic:\[ v = \sqrt{\frac{31.5}{0.0566}} = \sqrt{556.5401} \approx 23.59 \text{ m/s} \]

Key concepts

Physics Problem Solving

When tackling physics problems, a systematic approach can help you break down complex concepts into manageable steps. Start by understanding the given information and what is being asked. In our case, it's essential to determine the speed of the iron ball based on its kinetic energy.

Breaking it down:

• Identify what you're solving for: the speed (in meters per second).
• Note the given values: kinetic energy and mass.
• Understand the relationships involved, such as the equation connecting kinetic energy and velocity.

By addressing each part one step at a time, you can connect the dots to arrive at the solution.

Energy Equations

Energy equations help us understand how energy is expressed and conserved in different forms. For kinetic energy, we use the equation
\[ KE = \frac{1}{2}mv^2 \] This equation tells us that kinetic energy depends on both mass and velocity.

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

To find the velocity, you rearrange the formula to solve for \( v \):
\[ v = \sqrt{\frac{2KE}{m}} \]
This approach shows how changes in mass or energy affect speed.

Unit Conversion

Unit conversion is a key skill in physics to ensure all quantities are in the correct form before calculations. Often, you need to convert mass from grams to kilograms since energy equations require mass in kilograms. To do this, divide the mass in grams by 1000. For example,
\[ m = \frac{56.6}{1000} = 0.0566 \text{ kg} \] This conversion is crucial for accuracy in your calculations.

Remember:

• 1 kg = 1000 grams
• Always convert to the standard unit of measure used in equations

Effective unit conversion helps maintain consistency and prevent errors in your problem-solving process.