To push a stalled car, you apply a \(470-\mathrm{N}\) force at \(17^{\circ}\) to the car's motion, doing \(860 \mathrm{J}\) of work in the process. How far do you push the car?

Understanding the work-energy principle is crucial for solving a multitude of physics problems. In essence, the principle connects the work done by forces on an object to the object's kinetic energy. In the context of our physics work problem, work (\( W \) is defined as the force applied to an object (\( F \) multiplied by the distance (\( d \) that object moves as a result of that force, in the direction of the force.

The principle is mathematically expressed as \( W = F \times d \times \text{cos}(\theta) \) where \( \theta \) is the angle between the force and the direction of the motion. Net work done on an object results in a change in the object's kinetic energy — this forms the basis of the work-energy principle. If you pushed a car and did 860 J of work, according to the work-energy principle, you've essentially transferred 860 J of energy to the car, potentially increasing its kinetic energy by that amount.

The relationship between force and distance in the context of work can be described by a simple formula: \( W = F \times d \) when the force is applied in the direction of the motion. However, when the force is applied at an angle, as in our exercise, the distance calculation requires the inclusion of the cosine of the angle to account for the directional component.

The cosine factor adjusts the force to only include the portion acting in the direction of the motion. To solve for distance, we isolate \( d \) in the work formula: \( d = \frac{W}{F \times \text{cos}(\theta)} \). This rearrangement allows for direct calculation of how far the car moves when a certain amount of work is performed along a particular direction.

When dealing with angles in physics, particularly in relation to the circular motion (or components thereof), angles are often required to be in radians rather than degrees. To convert degrees to radians, the formula \( 1 \text{ rad} = \frac{180}{\text{π}} \text{ degrees} \) is used.

To perform the conversion: Take the angle in degrees and multiply it by \( \frac{\text{π}}{180} \). For example, in our car pushing problem, the angle of \( 17^{\text{°}} \) becomes \(\theta = 17 \times \frac{\text{π}}{180}\) radians. This step is vital before substitution into the work formula as trigonometric functions in physics formulas are implicitly assumed to take angles in radians.

The cosine function plays an important role in physics when it comes to understanding the component of a vector in the direction of another vector. In this context, cosine helps calculate the effective component of the force that is doing work in the direction of the displacement.

In our particular scenario, the use of the cosine function (\( \text{cos} \) arises because the force is not applied directly along the path of the car’s motion. Hence, we multiply the force by \( \text{cos}(17^{\text{°}}) \) to determine the component of the force that is parallel to the motion. It’s important to perform this step accurately to ensure the right force component is considered when calculating work or distance.