Question

A car of mass \(987 \mathrm{~kg}\) is traveling on a horizontal segment of a freeway with a speed of 64.5 mph. Suddenly, the driver has to hit the brakes hard to try to avoid an accident up ahead. The car does not have an ABS (antilock braking system), and the wheels lock, causing the car to slide some distance before it is brought to a stop by the friction force between its tires and the road surface. The coefficient of kinetic friction is \(0.301 .\) How much mechanical energy is lost to heat in this process?

Short answer

Answer: The mechanical energy lost to heat in this process is 399,843.5 Joules.

Step-by-step solution

1. Convert speed from mph to m/s

We need to convert the speed of the car from miles per hour (mph) to meters per second (m/s). To do this, we can use the following conversion factors: 1 mile = 1,609.34 meters 1 hour = 3,600 seconds So, the conversion factor from mph to m/s is: 1 mph = (1,609.34 m / 1 mile) × (1 hour / 3,600 s) Now, we can convert the speed of the car: Speed (m/s) = 64.5 mph × 1.60934 km/mile × (1000 m/km) / 3600 s/hour Speed (m/s) = 28.8 m/s

2. Calculate the initial kinetic energy

The kinetic energy (KE) of the car can be calculated using the formula: KE = (1/2) m v^2 where m is the mass of the car (987 kg) and v is its speed (28.8 m/s). KE = (1/2) × 987 kg × (28.8 m/s)^2 KE = 399,843.5 J (Joules)

3. Calculate the work done by friction force

To calculate the work done by friction force to stop the car, we can use the formula: W = μ × m × g × d where W is the work done, µ is the coefficient of kinetic friction (0.301), m is the mass of the car (987 kg), g is the acceleration due to gravity (9.8 m/s²), and d is the distance the car slides to stop. First, we need to find the distance (d) the car slides. The work-energy principle states that the work done on an object equals the change in its kinetic energy. So, the work done by friction to stop the car is equal to the initial kinetic energy: W = 399,843.5 J Now, we can solve for the distance (d): d = W / (μ × m × g) d = 399,843.5 J / (0.301 × 987 kg × 9.8 m/s²) d = 139.5 m

4. Find the mechanical energy lost to heat

The mechanical energy lost to heat is equal to the work done by friction: Mechanical energy lost to heat = Work done by friction Mechanical energy lost to heat = 399,843.5 J Therefore, the mechanical energy lost to heat in this process is 399,843.5 Joules.

Key concepts

Kinetic Energy

Kinetic energy represents the energy that an object possesses due to its motion. It is a form of energy that can be calculated for any moving object, from subatomic particles to large celestial bodies. When we talk about a moving car, like in our exercise, the kinetic energy is a crucial concept.

To calculate kinetic energy (\textbf{KE}), we use the formula \[\begin{equation}KE = \frac{1}{2}mv^2\text{,}\end{equation}\]where \[\begin{equation}m\end{equation}\]is the object's mass and \[\begin{equation}v\end{equation}\]is its velocity. In terms of units, kinetic energy is measured in Joules (J). For our example, the car's kinetic energy before it stops is equivalent to the work that must be done by the friction force to bring the car to rest—a force which in this case turns mechanical energy into heat.

Coefficient of Kinetic Friction

The coefficient of kinetic friction, denoted as \[\begin{equation}\mu\end{equation}\], is a dimensionless factor that describes the friction between two surfaces in relative motion. It is a measure of how easily one surface slides over another and determines the force of friction relative to the normal force (the force perpendicular to the contact surface).

Given a particular pair of surfaces, the coefficient of kinetic friction is generally a constant value under similar conditions. In the scenario of our car, the coefficient of kinetic friction between the locked tires and the road surface is given as 0.301. This value is used to calculate the work done by friction force and consequently, the mechanical energy transformation during the car's deceleration. It's essential to understand that the coefficient of kinetic friction plays a pivotal role in determining the deceleration rate and the distance over which the car slides.

Work-Energy Principle

The work-energy principle is a fundamental concept in classical mechanics, stating that the work done by all forces acting on an object will result in an equal change in the object's kinetic energy. This principle bridges the gap between force (work) and energy, allowing us to analyze various mechanical problems efficiently.

Mathematically, if the only significant force doing work is due to friction, such as in our textbook problem, the work (\[\begin{equation}W\end{equation}\]) done by friction force is the negative of the change in kinetic energy, because the kinetic energy decreases as the object comes to a stop. The calculation of work done by friction, therefore, can be found using the kinetic energy calculated before and after the event, which leads to the equation \[\begin{equation}W = \mu \times m \times g \times d\end{equation}\]. When applied, this principle tells us precisely how much mechanical energy has been transformed into heat due to the sliding friction between the car's tires and the road.