Question

A \(9.50-\mathrm{g}\) bullet has a speed of \(1.30 \mathrm{~km} / \mathrm{s}\). What is the kinetic energy of the bullet?

Short answer

The kinetic energy of the bullet is 8027.5 joules.

Step-by-step solution

Convert Mass to Kilograms

The mass of the bullet is initially given as \(9.50\) grams. To find the kinetic energy, we need the mass in kilograms. We know that \(1\) kilogram equals \(1000\) grams. So, to convert grams to kilograms, we use the formula:\( \text{mass in kilograms} = \frac{\text{mass in grams}}{1000} \). Therefore, the mass in kilograms is \(9.50 \div 1000 = 0.0095\) kg.

Convert Speed to Meters per Second

The speed of the bullet is given as \(1.30\) kilometers per second. To use it in the kinetic energy formula, we need it in meters per second. Since \(1\) kilometer equals \(1000\) meters, we multiply the speed by \(1000\) to convert to meters per second: \(1.30 \times 1000 = 1300\) m/s.

Apply the Kinetic Energy Formula

The formula for kinetic energy (KE) is \( \text{KE} = \frac{1}{2}mv^2 \), where \(m\) is the mass in kilograms and \(v\) is the velocity in meters per second. Substitute the values we have calculated: \( m = 0.0095 \text{ kg} \) and \( v = 1300 \text{ m/s} \). So, \(\text{KE} = \frac{1}{2} \times 0.0095 \times 1300^2 \).

Calculate the Kinetic Energy

Calculate \(1300^2 = 1690000\). Now substitute into the equation: \(\text{KE} = \frac{1}{2} \times 0.0095 \times 1690000 \). First, multiply \(0.0095 \times 1690000 = 16055\). Then, calculate \(\frac{1}{2} \times 16055 = 8027.5\). The kinetic energy of the bullet is \(8027.5\) joules.

Key concepts

Mass Conversion

To calculate kinetic energy, one essential step is converting mass into the proper units, which are kilograms in the International System of Units (SI). The mass of an object affects its kinetic energy significantly, so it's crucial to ensure our units are correct. In many physics problems, such as this one, mass may be given in grams. For easy manipulation in formulas, we need to convert these grams to kilograms. This conversion is quite simple:

• Remember that 1 kilogram equals 1000 grams.
• To find the mass in kilograms, you divide the given mass in grams by 1000.

Applying this process, a bullet with a mass of 9.50 grams is converted to 0.0095 kilograms (\[\text{mass in kilograms} = \frac{9.50}{1000} = 0.0095 \text{ kg}\]). This step ensures the mass is appropriately scaled for further calculations.

Speed Conversion

Kinetic energy calculations require the velocity to be in meters per second (m/s). Often, speed might be given in other units, like kilometers per second (km/s). To make the calculations work smoothly and to adhere to SI units, we need to convert these units using a simple conversion:

• 1 kilometer equals 1000 meters.
• Multiply the speed in kilometers per second by 1000 to convert it to meters per second.

Using this method, a bullet traveling at 1.30 km/s is found to have a speed of 1300 m/s (\[1.30 \times 1000 = 1300\text{ m/s} \]). Appropriately converted speed ensures accurate results in the kinetic energy calculation.

Formula Application

Once you have your mass in kilograms and speed in meters per second, you can apply the kinetic energy formula. The formula for kinetic energy is:\[\text{KE} = \frac{1}{2} mv^2\]where:

• KE is the kinetic energy,
• m is the mass in kilograms,
• v is the velocity in meters per second.

Substitute the values you've calculated:

• Mass m = 0.0095 kg
• Velocity v = 1300 m/s

So your equation becomes:\[\text{KE} = \frac{1}{2} \times 0.0095 \times 1300^2\]By squaring the velocity, multiplying by the mass, and then the entire product by 0.5, you find the kinetic energy, which in this scenario comes out to be 8027.5 joules.

Potential Energy

While this exercise focuses on kinetic energy, it's useful to understand its counterpart, potential energy. Potential energy is stored energy based on an object's position rather than its motion. The most common type is gravitational potential energy, especially relevant when objects are elevated from the ground. This type of potential energy is calculated using the formula:\[\text{PE} = mgh\]where:

• m is mass (in kilograms),
• g is the acceleration due to gravity (approximately 9.81 m/s² on Earth),
• h is the height from a datum (in meters).

While kinetic energy is related to speed, potential energy comes into play when considering how position affects an object's ability to do work. In a complete energy analysis, both kinetic and potential energy components give insights into the behavior of objects in motion and at rest.