Question

A car of mass \(m=1250 \mathrm{~kg}\) is traveling at a speed of \(v_{0}=105 \mathrm{~km} / \mathrm{h}(29.2 \mathrm{~m} / \mathrm{s}) .\) Calculate the work that must be done by the brakes to completely stop the car.

Short answer

Short Answer: To find the work done by the brakes to stop a 1250 kg car with an initial speed of 29.2 m/s, we first calculated the initial kinetic energy (KE₀) = 528,020 J and final kinetic energy (KE₁) = 0 J. Then, we found the change in kinetic energy (ΔKE) = -528,020 J. The work done by the brakes, according to the work-energy principle, is equal to the change in kinetic energy, which is -528,020 J.

Step-by-step solution

Identify the formula to use

In this case, we need to use the work-energy principle formula, which is: \[W = ΔKE\] Where W is the work done, and ΔKE is the change in kinetic energy of the car.

Calculate the initial kinetic energy

The kinetic energy (KE) can be calculated using the formula: \[KE = \frac{1}{2}mv^2\] Where m is the mass of the car and v is its speed. We know that the mass (m) is 1250 kg, and the initial speed (v₀) is 29.2 m/s. So, the initial kinetic energy (KE₀) can be calculated as follows: \[KE_0 = \frac{1}{2}(1250 \mathrm{~kg})(29.2 \mathrm{~m} / \mathrm{s})^2\]

Calculate the final kinetic energy

The car comes to a complete stop, so its final speed (v₁) is 0. Calculate the final kinetic energy (KE₁) using the same formula but with v₁ = 0. \[KE_1 = \frac{1}{2}(1250 \mathrm{~kg})(0 \mathrm{~m} / \mathrm{s})^2\]

Calculate the change in kinetic energy

Now that we have the initial and final kinetic energies, we can calculate the change in kinetic energy (ΔKE): \[ΔKE = KE_1 - KE_0\]

Calculate the work done by the brakes

Finally, we can calculate the work done by the brakes using the work-energy principle formula from step 1: \[W = ΔKE\] Now plug in the values calculated in the previous steps and solve for the work done by the brakes.

Key concepts

Kinetic Energy

Understanding kinetic energy is pivotal to solving many physics problems involving motion. It's the energy a body possesses due to its movement. For an object with mass m traveling at a velocity v, the kinetic energy (KE) is given by the equation

\[\begin{equation}KE = \frac{1}{2}mv^2\end{equation}\]
Here, \( \frac{1}{2} \) is a constant factor, m represents the mass of the object, and v is its velocity. The unit of kinetic energy is the Joule (J), which is the same as kg\(\cdot\)m\(^2\)/s\(^2\).

Why Kinetic Energy is Important

Kinetic energy plays a fundamental role in the work-energy principle. When a car moves, it has kinetic energy proportional to its speed squared. If we want to bring the car to a halt, like in the problem above, we need to remove this energy from the car. This understanding helps us determine the effort or work required to stop moving objects, which is especially useful in safety engineering and vehicle design.

Work Done by Brakes

The work done by brakes is a classic example of energy transformation. Brakes stop a car by converting the vehicle's kinetic energy into heat through friction. The work done by the brakes, in this context, equals the amount of kinetic energy the car initially had.

To find the work (W) done by the brakes, we use the work-energy principle:

\[\begin{equation}W = \Delta KE = KE_{\text{initial}} - KE_{\text{final}}\end{equation}\]
where \( \Delta KE \) is the change in kinetic energy. When a car comes to a complete stop, its final kinetic energy is zero. Hence, the work done by the brakes is equal to the initial kinetic energy of the car.

Understanding Brake Efficiency

In this way, analyzing the work done by brakes helps us understand brake efficiency and the heat dissipation required to prevent brake failure. Knowing this is crucial for vehicle safety and performance tuning.

Physics Problem Solving

Physics problem solving often involves a step-by-step approach to break down complex scenarios into manageable calculations. The problem at hand showcases a methodical way to apply the work-energy principle effectively.

Firstly, clearly identify and understand the known quantities and what is being asked. For example, discerne the mass of the car and its initial velocity. Then, utilize the correct physics equations related to the concept, such as the kinetic energy formula.

• Identify the relevant formulas and concepts
• Calculate initial conditions, like kinetic energy before braking
• Determine final conditions, which is typically zero kinetic energy for a stopped object
• Resolve the necessary changes, in this case, change in kinetic energy
• Apply the found value to find the work done, hence completing the problem

Through this structured approach, students can manage a wide array of problems in physics by altering the process to fit different scenarios. Both the understanding of key concepts and the methodical attitude to problem-solving play crucial roles in successfully navigating physics exercises.