Problem 96 - Physics for Scientists and Engineers with Modern Physics

Question

Proper design of automobile braking systems must account for heat buildup under heavy braking. Calculate the thermal energy dissipated from brakes in a \(1500-\mathrm{kg}\) car that descends a \(17^{\circ}\) hill. The car begins braking when its speed is \(95 \mathrm{~km} / \mathrm{h}\) and slows to a speed of \(35 \mathrm{~km} / \mathrm{h}\) in a distance of \(0.30 \mathrm{~km}\) measured along the road.

Step-by-Step Solution

Answer

The thermal energy dissipated from the brakes is approximately 451,396 J.

Step 1: Convert Speeds to Meters per Second

First, convert the initial and final speeds from kilometers per hour to meters per second.Initial speed: \\[ 95 \text{ km/h} = \frac{95 \times 1000}{3600} \approx 26.39 \text{ m/s} \]Final speed: \\[ 35 \text{ km/h} = \frac{35 \times 1000}{3600} \approx 9.72 \text{ m/s} \]

Step 2: Calculate the Change in Kinetic Energy

Calculate the initial and final kinetic energies using the formula \( KE = \frac{1}{2}mv^2 \):Initial kinetic energy: \[ KE_{i} = \frac{1}{2} \times 1500 \times (26.39)^2 \approx 522,280.65 \text{ J} \]Final kinetic energy: \[ KE_{f} = \frac{1}{2} \times 1500 \times (9.72)^2 \approx 70,884.60 \text{ J} \]The change in kinetic energy \( \Delta KE \) is:\[ \Delta KE = KE_{i} - KE_{f} = 522,280.65 - 70,884.60 = 451,396.05 \text{ J} \]

Step 3: Consider Energy Lost to Heat

The energy dissipated as heat through the brakes is equal to the change in kinetic energy because the car slows down only due to braking and as a result of converting kinetic energy to thermal energy.So, the thermal energy dissipated through the brakes is \( 451,396.05 \text{ J} \).

Key Concepts

Thermal Energy

When braking a vehicle, the kinetic energy that propels the car is transformed into another form of energy called "thermal energy." This transformation is crucial in stopping a car effectively without causing damage to the braking system.
During braking, friction is created between the brake pads and the brake rotors. This friction heats up the materials and leads to energy loss in the form of heat.
The car's brakes must dissipate this thermal energy efficiently to avoid overheating which can compromise the braking system's reliability.

• Efficient dissipation of thermal energy prevents brake fade, a condition where the brakes become less effective due to excessive heat.
• The material and design of brake components are critical in managing thermal energy.

Understanding how thermal energy works in brakes can help in designing better braking systems that are both safe and durable.

Kinetic Energy

Kinetic energy is the energy of motion. In the context of automobiles, it refers to the energy that a moving vehicle carries due to its velocity and mass. A heavier or faster vehicle has more kinetic energy.
To compute the kinetic energy (\[KE\]) of an automobile, the formula used is:\[ \text{KE} = \frac{1}{2} m v^2 \]where \( m \) is the mass of the vehicle and \( v \) is the speed of the car.
In the exercise, the car's initial and final kinetic energies were calculated using this formula. By knowing the change in kinetic energy during the braking process, we can determine how much energy needs to be transformed or dissipated.

• Initial kinetic energy is always greater than the final kinetic energy when a car is braking.
• Reducing the car's speed translates to a reduction in kinetic energy.

Understanding kinetic energy aids in assessing how energy must be managed by the brake system during vehicle deceleration.

Energy Dissipation

Energy dissipation in the context of braking refers to the process of losing energy, mostly in the form of heat. This process is crucial in slowing down a car safely.
When a vehicle brakes, it converts kinetic energy to thermal energy. Essentially, the energy that was used to keep the car moving is now being removed via friction as heat in the brakes.
Here's a simple breakdown:

• As the vehicle slows down, energy is dissipated through brake pads and rotors.
• Efficient energy dissipation is vital to preventing heat build-up that can lead to brake failure.
• The design of braking systems incorporates materials that can cope with high temperatures and repeated use.

By managing energy dissipation effectively, brakes remain functional and safe even under heavy and repeated use, ensuring longer lifespan and reliability of the vehicle's braking system.
Understanding energy dissipation helps in evaluating how well a braking system will perform in real-world conditions.