At what speed must a 950 -kg subcompact car be moving to have the same kinetic energy as a \(3.2 \times 10^{4}-\mathrm{kg}\) truck going \(20 \mathrm{km} / \mathrm{h} ?\)

Understanding the kinetic energy (KE) formula is essential to solving problems related to motion.
The basic formula for kinetic energy is
\[\begin{equation} KE = 0.5 \times m \times v^2 \end{equation}\]
where 'm' represents the mass of the object and 'v' is its velocity. Kinetic energy tells us how much work an object can do by virtue of its motion.
It is the energy that it would transfer to an object upon collision, or it would need to be removed to bring the object to rest. This is why heavier vehicles like trucks possess greater kinetic energy than smaller cars at the same speed. When applying the formula, it is critical to ensure that the mass is in kilograms (kg) and the velocity is in meters per second (m/s), which leads us seamlessly into the importance of unit conversion in physics calculations.

In physics problems, consistency in units is key. A common issue students face is how to convert units from one system to another, especially in kinetic energy calculations.
In the exercise, for instance, the given truck's speed was initially in kilometers per hour (km/h), which is not compatible with the SI units used in the kinetic energy formula—meters per second (m/s).
The conversion process here employed the fact that there are 1000 meters in a kilometer and 3600 seconds in an hour. So, the conversion formula becomes
\[\begin{equation} v (\text{m/s}) = v (\text{km/h}) \times \frac{1000 \text{ m/km}}{3600 \text{ s/h}} \end{equation}\]
Understanding and applying the correct unit conversions is crucial for accurate physics calculations. These conversions often involve multiplying or dividing by conversion factors, as shown in the solution. Failing to convert units properly can lead to incorrect results and misconstrued understanding of the physical concepts at play.

The principle of conservation of energy is a fundamental concept in physics, stating that energy cannot be created or destroyed, only transformed from one form to another.
In the context of the exercise, we've assumed that the kinetic energies of both the truck and the car should be equal - which means that if one were to transfer its energy to another, the motion imparted would result in the same kinetic energy.
In real-world situations, this principle allows us to solve for unknown quantities by setting kinetic energy equal before and after an event, assuming no energy is lost to other forms like heat or sound.
For the car to have the same kinetic energy as the truck, we needed to determine at what speed the car would have to move. This is a direct application of the conservation of energy, demonstrating how energy's constancy is a powerful tool in problem-solving. Each form of energy can be converted into another - potential energy can become kinetic, electric to thermal, and so forth - but the total amount within a closed system remains constant.