Question

An automobile with passengers has a weight of \(16,400 \mathrm{~N}\) \((=3680 \mathrm{lb})\) and is moving up a \(10^{\circ}\) slope with an initial speed of \(70 \mathrm{mi} / \mathrm{h}(=113 \mathrm{~km} / \mathrm{h})\) when the driver begins to apply the brakes. The car comes to a stop after traveling \(225 \mathrm{~m}\) along the inclined road. Calculate the work done by the brakes in stopping the car, assuming that all other energy transfers in this problem (such as heat and internal energy) can be neglected.-

Short answer

The work done by the brakes in Joules would be calculated by the difference between the initial kinetic energy and potential energy. It may come out to be negative which signifies that the work is done against the motion of the car.

Step-by-step solution

Convert Units

Convert initial speed from miles per hour to meters per second by multiplying it by the conversion factor of 0.44704. You get \(70 \mathrm{mi} / \mathrm{h} * 0.44704 = 31.29 \mathrm{m/s} \)

Calculate Initial Kinetic Energy

Calculate the initial kinetic energy of the car using the formula: \( KE = 0.5 * m * v^2 \) where m is the mass of the car and v is the velocity. But since the weight of the car is given, we first need to calculate the mass using the formula: \( m = F/g \) where F is the force (equal to the weight of the car) and g is the gravitational acceleration (= 9.8 m/s^2). After calculating the mass, use it to calculate the initial kinetic energy.

Calculate Potential Energy

The potential energy (PE) of the car when it's at the top of the inclined path can be calculated using the formula: \( PE = m * g * h \), where m is the mass of the car, g is the gravity and h is the height of the inclined road. We can calculate the height using the formula: \( h = s * sin(\Theta) \) where s is the distance the car moves along the incline (= 225m) and \( \Theta \) is the angle of the incline (= 10 degrees). After calculating the height, use it to calculate the potential energy.

Calculate Work Done

The total mechanical energy of the system before the brakes are applied will be the kinetic energy. The total mechanical energy after the car stops will be the potential energy. The work done by the brakes is the difference between the two. So, \( Work = KE - PE \)

Key concepts

Kinetic Energy

Kinetic energy (KE) is the energy an object possesses due to its motion. It depends on the mass of the object and the square of its velocity (speed). The formula to calculate kinetic energy is given as:
\[ KE = 0.5 \times m \times v^2 \] Where:\(m\) is the mass of the object (in kilograms), and \(v\) is the velocity (in meters per second).
In the context of the exercise, the car's initial kinetic energy is significant because it represents the amount of energy that needs to be dissipated by the brakes to bring the car to a stop. When the brakes are applied, they do work against the car's kinetic energy, converting it into other forms of energy, like heat, due to friction and thereby slowing down the car.

Potential Energy

Potential energy (PE) is the energy held by an object because of its position relative to other objects, associated with its height above ground. The formula for gravitational potential energy is:
\[ PE = m \times g \times h \] Where:\(m\) is the mass,\(g\) is the acceleration due to gravity (9.8 m/s^2 on Earth), and \(h\) is the height of the object above the ground level.
The exercise references potential energy to determine how much energy the car has when it stops at a certain height after traveling up the incline. This energy could potentially be converted back into kinetic energy if the car were to roll back down the slope.

Mechanical Energy Conservation

The principle of mechanical energy conservation states that in an isolated system (where there's no external work done), the total mechanical energy (the sum of kinetic and potential energy) remains constant. Therefore, the energy can change forms—such as from kinetic to potential and vice versa—but the total amount of mechanical energy is conserved.
In a scenario such as a car braking on an incline, mechanical energy is not conserved because the brakes apply an external force, converting mechanical energy to heat. However, understanding this principle is crucial because it allows us to analyze the situation before the brakes are applied. At that moment, we can calculate the car's total mechanical energy by adding the kinetic and potential energies.

Inclined Plane Physics

Inclined planes simplify the analysis of forces and work done on an object moving on a slope. The physics of an inclined plane consider gravity's effect on the object in two components: one perpendicular to the plane and the other parallel to it.
The height \(h\) of an inclined plane is related to the length of the slope \(s\) and the angle of inclination \(\theta\) using the trigonometric relation:
\[ h = s \times \sin(\theta) \] This relation is key for calculating the potential energy change as an object moves up or down an incline. In the exercise, this formula helps us calculate the height gained by the car, which, multiplied by the car's mass and the acceleration due to gravity, gives us the potential energy when the car is at a stop on the incline.