Question

A car does the work \(W_{\text {car }}=7.0 \cdot 10^{4} \mathrm{~J}\) in traveling a distance \(x=2.8 \mathrm{~km}\) at constant speed. Calculate the average force \(F\) (from all sources) acting on the car in this process.

Short answer

Answer: The average force acting on the car is 25 N.

Step-by-step solution

Convert distance to meters

We are given the distance traveled by the car in kilometers (\(x=2.8\,\text{km}\)). To solve the problem, we need the distance in meters. To convert kilometers to meters, multiply the distance by 1000: \(x = 2.8\,\text{km} \times 1000 = 2800\,\text{m}\)

Use the work-energy theorem

Since the car is moving at a constant speed, there is no change in kinetic energy, and we can use the equation for work done, \(W=F \cdot d \cdot \cos \theta\). Since we assume the angle \(\theta\) to be \(0^{\circ}\), we have \(\cos \theta = 1\). Thus, the equation simplifies to: \(W = F \cdot d\)

Solve for the average force

Now, we have all the information needed to solve for the average force \(F\). We know the work done by the car is \(W_{\text{car}} = 7.0 \cdot 10^4 J\), and the distance traveled is \(x=2800 m\). Using the equation \(W=F \cdot d\), we can solve for the force: \(F = \frac{W}{d} = \frac{7.0 \cdot 10^4\,\text{J}}{2800\,\text{m}}\)

Calculate the average force

Now, perform the calculation to find the average force acting on the car: \(F = \frac{7.0 \cdot 10^4\,\text{J}}{2800\,\text{m}} = 25\,\text{N}\) Thus, the average force acting on the car during this process is \(F = 25\,\text{N}\).

Key concepts

Work-Energy Theorem

The work-energy theorem is a fundamental principle in physics that connects the concept of work with the change in kinetic energy of an object. According to this theorem, the work done on an object is equal to the change in its kinetic energy. In the context of the given problem, since the car is moving at a constant speed, there is no change in kinetic energy, meaning any work done is related to overcoming external forces, such as friction or air resistance.
Understanding that the work done by the car, which is given as 70,000 Joules, is used to calculate the impact of these forces is crucial. When a constant speed is maintained, the work accounts for forces that are resisting the motion of the car, thus enabling the calculation of average force applied. This concept highlights the importance of balancing forces when working in dynamic systems to maintain equilibrium.

Average Force Calculation

Calculating the average force acting on an object can be simplified by using the work-energy relationship. In this problem, the car is moving at a constant speed. Therefore, the average force can be calculated using the equation for work:

• Work ( \( W \) ) equals the force ( \( F \) ) times the distance ( \( d \) ).

Given the work (\( W = 70,000 \, \text{Joules} \)) and the distance traveled (\( d = 2800 \, \text{meters} \)), we find the average force (\( F \)) by rearranging the formula:

• \( F = \frac{W}{d} = \frac{70,000 \, \text{J}}{2800 \, \text{m}} = 25 \, \text{N} \)

This result means that the average force needed to keep the car moving at a consistent speed, overcoming any forces like friction, is 25 Newtons. This calculation illustrates how forces are distributed over a distance to achieve work, central to many engineering and physical scenarios.

Unit Conversion in Physics

Unit conversion is an important skill in physics, allowing quantities to be expressed in suitable units for calculations and analysis. In this particular exercise, distance was originally provided in kilometers, necessitating conversion to meters for easier application in the work formula.
It's essential to remember the conversion factor: 1 kilometer equals 1000 meters. This is a basic yet crucial step in ensuring accuracy across physics problems. By converting properly:

• \( x = 2.8 \, \text{km} \times 1000 = 2800 \, \text{m} \)

Now, with the distance in meters, the calculation of forces and work becomes consistent with standard physics units, such as Joules for work and Newtons for force.
This highlights the significance of unit conversion, ensuring that all aspects of a problem align tidily, facilitating accurate and meaningful results.