Question

A limo is moving at a speed of \(100 . \mathrm{km} / \mathrm{h}\). If the mass of the limo, including passengers, is \(1900 . \mathrm{kg}\), what is its kinetic energy?

Short answer

Answer: The kinetic energy of the limo is 734,250 Joules.

Step-by-step solution

Convert the speed unit

Given that the limo is moving at a speed of \(100\; km/h\), we need to convert this speed from kilometers per hour (km/h) to meters per second (m/s). To do this, we can use the conversion factor: \(1\; km/h = \frac{1000\; m}{3600\; s}\) So, the speed of limo in meters per second (m/s) is: \(v = 100 \times \frac{1000}{3600}\; m/s = \frac{100000}{3600}\; m/s = 27.78\; m/s\)

Apply the kinetic energy formula

Now, we have the mass of the limo \(m = 1900\; kg\) and its speed \(v = 27.78\; m/s\). Using the kinetic energy formula, we can compute the kinetic energy of the limo: \(KE = \frac{1}{2}mv^2\)

Calculate the kinetic energy

Plug in the values for mass and speed into the formula: \(KE = \frac{1}{2}(1900\; kg)(27.78\; m/s)^2\) To get the kinetic energy: \(KE = 0.5 \times 1900 \times 27.78^2 = 0.5 \times 1900 \times 772.3684 = 734250\; J\) Therefore, the kinetic energy of the limo is \(734250\; Joules\).

Key concepts

Kinetic Energy Formula

Kinetic energy, denoted as KE, represents the energy that an object possesses due to its motion. It quantifies how much work an object can do by virtue of its velocity. The kinetic energy formula is given by:
\[\begin{equation}KE = \frac{1}{2}mv^2\end{equation}\]
where m is the mass of the object in kilograms (kg), and v is its velocity in meters per second (m/s). This equation shows that the kinetic energy of an object is directly proportional to its mass and the square of its velocity, meaning that an increase in either the mass or velocity leads to a more substantial increase in kinetic energy.

Unit Conversion

Unit conversion is critical in physics as it ensures that equations are dimensionally consistent, and the final results are in the correct units. For instance, in kinetic energy calculations, speed must be in meters per second rather than kilometers per hour.
To convert km/h to m/s, we use the relationship:
\[\begin{equation}1 \text{ km/h} = \frac{1000 \text{ m}}{3600 \text{ s}}\end{equation}\]
Therefore, a simple multiplication of the speed value by this conversion factor translates the velocity into the appropriate units. Understanding unit conversion allows for the accurate application of formulas and the correct interpretation of results.

Physics Problem Solving

Solving physics problems often involves a systematic approach, which includes understanding the problem, identifying the relevant principles, applying mathematical relationships, and then carrying out the computations.

Understanding the Problem

Start by comprehending what is being asked and identify the given information. In our limo example, we need to find the kinetic energy, given the mass and speed of the limo.

Identifying Relevant Principles

Determine the physical laws or formulas that apply to the scenario. Here, we utilize kinetic energy's definition and formula.

Applying Mathematical Relationships

Use algebraic manipulation and unit conversions as needed to solve for the unknowns. This involves converting speeds and substituting the values into the kinetic energy formula.

Executing Computations

Finally, perform the calculations carefully, taking into account significant figures and unit consistency, to reach the answer. Being thorough at each step ensures a correct solution, as demonstrated with our limo's kinetic energy problem.
Problem solving in physics not only reinforces conceptual understanding but also hones mathematical skills, both critical for success in the field.