Question

A \(1.50 \cdot 10^{3}-\mathrm{kg}\) car travels \(2.50 \mathrm{~km}\) up an incline at constant velocity. The incline has an angle of \(3.00^{\circ}\) with respect to the horizontal. What is the change in the car's potential energy? What is the net work done on the car?

Short answer

Answer: The change in the car's potential energy is approximately \(1.92 \times 10^6\ \text{J}\), and the net work done on the car is also approximately \(1.92 \times 10^6\ \text{J}\).

Step-by-step solution

Convert the distance to meters

To find the potential energy change, we need to work in standard units. So, convert the distance the car travels in kilometers to meters: \(2.5 \ \text{km} = 2500\ \text{m}\)

Calculate the height gained

Now, we must calculate the height gained while traversing the inclined plane. We can use the sine function to find the height gained. We have the equation: \(h = d \cdot \sin{\theta}\) where \(h\) is the height, \(d\) is the distance traveled, and \(\theta\) is the angle of the incline. Plug the given values into the equation: \(h = 2500\ \text{m} \cdot \sin{3^\circ}\) \(h = 2500\ \text{m} \cdot 0.0523\) \(h \approx 130.66\ \text{m}\)

Calculate the change in potential energy

Now we can calculate the change in potential energy using the formula, \(PE = mgh\), where \(m\) is the mass, \(g\) is the acceleration due to gravity \(9.81\ \text{m/s}^2\), and \(h\) is the height. \(\Delta PE = mgh\) \(\Delta PE = 1.50 \times 10^3\ \text{kg} \cdot 9.81\ \text{m/s}^2 \cdot 130.66\ \text{m}\) \(\Delta PE \approx 1.92 \times 10^6\ \text{J}\)

Calculate the net work done on the car

Since the car moves at a constant velocity, the net work done on the car should be equal to the change in potential energy. \(W_{net} = \Delta PE\) \(W_{net} \approx 1.92 \times 10^6\ \text{J}\) So, the change in the car's potential energy is about \(1.92 \times 10^6\ \text{J}\), and the net work done on the car is also approximately \(1.92 \times 10^6\ \text{J}\).

Key concepts

Inclined Plane

The inclined plane is a classic physics problem that brings to life the concepts of force, motion, and energy. By analyzing an object moving along an inclined plane, students can better understand how these physical principles interact.

An inclined plane is essentially a flat surface that is tilted at an angle, in this case, a 3-degree angle, to the horizontal. This setup presents a scenario where the weight of the object causes it to move or exert force down the slope due to gravity. However, instead of simply falling straight down, the object moves along the slope, which means we need to consider both vertical and horizontal components of motion.

When a car moves up an inclined plane at a constant velocity, as in our exercise, it means that the net force along the plane is zero. However, this does not mean that no work is being done. The engine of the car is doing work to overcome gravitational pull and possibly frictional forces to maintain that constant velocity, a concept intricately tied to the work-energy principle.

Work-Energy Principle

The work-energy principle is a fundamental idea in physics that relates the work done on an object to the change in its energy. Work is defined as the transfer of energy, and in the case of our exercise, the net work done on the car by the engine's force is what causes the change in the car's potential energy.

The formula for work is given by the equation: \(W = Fd\), where \(W\) is work, \(F\) is force, and \(d\) is distance in the direction of the force. In the case of the car on the inclined plane, the force exerted by the engine moves the car upward against gravity over some distance, which in turn, increases the car's potential energy.

When an object moves at a constant velocity, as in our example, the total work done by all the forces (like the engine's force, gravitational force, and any frictional forces) must be zero, since the object's kinetic energy isn't changing. However, despite this balance, there's still a change in potential energy because the car is being lifted to a greater height, indicating that work is indeed being done.

Kinematics

Kinematics is the branch of physics that describes the motion of objects without necessarily dealing with the forces that cause them. It looks at variables such as displacement, velocity, and acceleration to narrate the story of an object's movement.

In our scenario, kinematics is used to determine important elements like displacement and height of the inclined plane. For instance, the exercise employs trigonometry and the kinematic concept of displacement to find the vertical height the car ascends as it moves up the slope. The itinerary of the car is known thanks to the distance traveled and the angle of the incline, leading us to compute this vertical component with sin(theta), an essential part of the kinematic toolbox.

Such calculations become the stepping stones to understanding more complex elements like energy changes. Kinematics doesn't concern itself with the forces involved (which would be the realm of dynamics) but is a critical part of analyzing motion, and in cases like these, it helps us bridge the gap to understanding the work-energy relationship in physical systems.