Question

Calculate the kinetic energies of the following objects moving at the given speeds: ( \(a\) ) a \(110-\mathrm{kg}\) football linebacker running at \(8.1 \mathrm{~m} / \mathrm{s} ;\) (b) a \(4.2-\mathrm{g}\) bullet at \(950 \mathrm{~m} / \mathrm{s} ;\) ( \(c\) ) the aircraft carrier Nimitz, 91,400 tons at \(32.0\) knots.

Short answer

The kinetic energy of the linebacker is approximately \(3614.5 \, J\), the bullet approximately \(1898.25 \, J\), and the aircraft carrier approximately \(2.287 \times 10^{11} \, J\).

Step-by-step solution

Calculate the Kinetic Energy of the Linebacker

First, convert the mass of the linebacker to kg, if not already in this unit. In this case, it is. Then, plug the values into the kinetic energy formula to solve for the linebacker's kinetic energy: \(KE = \frac{1}{2}mu^2 = \frac{1}{2}(110 \, kg)(8.1 \, m/s)^2\)

Calculate the Kinetic Energy of the Bullet

The mass of the bullet is given in grams, so it needs to be converted to kilograms by dividing by 1,000. After that, calculate the kinetic energy of the bullet by plugging the values into the kinetic energy formula: \(KE = \frac{1}{2}mu^2 = \frac{1}{2}(0.0042 \, kg)(950 \, m/s)^2.\)

Calculate the Kinetic Energy of the Aircraft Carrier

First, convert the mass of the aircraft carrier from tons to kilograms by multiplying by 1,000. The speed is given in knots, it needs to be converted to meters per second by multiplying by 0.51444 (since 1 knot = 0.51444 m/s). Now with values in the appropriate units, you can solve for the kinetic energy with the formula: \(KE = \frac{1}{2}mu^2 = \frac{1}{2}(91400000 \, kg)(32 \times 0.51444 \, m/s)^2\)

Key concepts

mechanics

Mechanics is a branch of physics that deals with the behavior of objects and the forces acting upon them. It includes the study of motion, forces, and energy. In the context of kinetic energy, mechanics helps us understand how the speed and mass of an object influence its energy.Kinetic energy is a form of mechanical energy that an object possesses due to its motion. It is calculated using the formula: \[KE = \frac{1}{2} m u^2\]where \(m\) is the mass of the object and \(u\) is its speed. This equation shows us that kinetic energy depends on both mass and the square of speed. Due to the squaring of velocity, speed has a more significant impact on kinetic energy than mass. For instance:

• A small increase in speed can lead to a substantial increase in kinetic energy.
• Doubling the speed of an object will quadruple its kinetic energy.

By applying mechanics to solve problems, we can calculate the kinetic energies of objects like linebackers, bullets, and aircraft carriers, each with unique masses and speeds.

energy conversion

Energy conversion is the process of transforming one form of energy into another. In everyday physics problems, such as those involving kinetic energy, we often convert stored potential energy into kinetic energy when objects are set in motion. When thinking about energy conversion, it's helpful to remember the law of conservation of energy. This law states that energy cannot be created or destroyed, only transformed from one form to another. Here are some key points:

• When an object moves, potential energy (such as gravitational energy) can be converted into kinetic energy.
• In collisions or movements, kinetic energy can also be transferred from one object to another.

For example, a linebacker running on the field converts the chemical energy from food into kinetic energy. Similarly, a bullet fired from a gun converts the stored potential energy from the gunpowder into kinetic energy as it flies toward its target.

object motion

Object motion is a fundamental concept in physics that describes how objects move over time due to various forces. Understanding object motion is crucial in calculating kinetic energy, as kinetic energy is directly related to the motion of objects. When dealing with motion, it’s important to consider both speed and direction, collectively known as velocity. A moving object, like a bullet or a running linebacker, gains kinetic energy because of its velocity. Here are a few things to remember:

• Speed and mass of an object determine its kinetic energy.
• In the absence of external forces, objects in motion stay in motion, as per Newton's First Law of Motion.
• Changes in velocity impact the kinetic energy greatly, due to its dependence on velocity squared in the calculation.

Converting units of mass and speed is also often necessary in physics problems, ensuring all calculations are accurate. For instance, converting tons to kilograms or knots to meters per second helps when calculating an aircraft carrier's kinetic energy.