91Ó°ÊÓ

In physics, vector components are crucial for understanding how a vector quantity, like force or displacement, can be broken down in terms of its direction. A vector can be split into parts along various axes, typically the x and y axes in two-dimensional space.
In our exercise, the car’s displacement direction is specified relative to the x-axis. This means we can determine the displacement vector’s components using trigonometric identities.
By converting the angle of the car’s travel direction, 240°, into familiar terms, we recognize it correlates with the third quadrant on the unit circle. Hence, both x and y components of displacement have negative values:

• The x-component moves left, given by the equation for displacement: \[d_x = 48.0 \cdot \cos(240.0^\circ) = -24.0 \, \text{m}\]
• The y-component moves downward, using:\[d_y = 48.0 \cdot \sin(240.0^\circ) = -41.57 \, \text{m}\]

This understanding allows us to evaluate how the force acts along those directions, by connecting the geometry with physics calculations.

The dot product, also known as scalar product, is an algebraic operation that takes two equal-length sequences of numbers—often coordinate vectors—and returns a single number. This result gives a measure of how two vectors are aligned.
In our scenario, calculating the dot product involves multiplying the components of the force with the respective components of the displacement. The formula is:
\[ W = \vec{F} \cdot \vec{d} = F_x \cdot d_x + F_y \cdot d_y \]
This operation helps us to determine the work done by the force on the car.
With numerical substitution, we end up performing these calculations:

• Multiplying corresponding x-components: \[68.0 \times 24.0 = 1632.0\]
• Multiplying y-components: \[36.0 \times 41.57 = 1496.52\]

The dot product becomes vital in determining that the overall work leads to energy added to the car.

Trigonometric identities are equations involving trigonometric functions that are true for any value of the occurring variables. These identities are useful tools for simplifying expressions in relation to angles.
In this exercise, we discovered how angles assist in detailing vector components:

• The cosine function for extracting the horizontal (x-direction) component of the car’s displacement:\[\cos(240.0^\circ) = -0.5\]
• The sine function for the vertical (y-direction) component:\[\sin(240.0^\circ) = -0.866\]

This approach allows us to conveniently break down vectors using angles, providing a clear link between geometry and physics. Without these identities, analyzing forces and motions in different directions would be notably formidable.

A constant force implies that the force applied on an object does not change in magnitude or direction over time. This creates a steady influence that simplifies calculations of work and energy, as variations due to fluctuating forces need not be considered.
In the exercise at hand, the force applied is given as a two-component vector:\((-68.0 \, \text{N})\hat{\imath} + (36.0 \, \text{N}) \hat{\jmath}\). Each component describes the constant horizontal and vertical portions of the force's influence.
As the force acts steadily over the 48 meters of travel, the application remains straightforward:

• The horizontal force opposes the displacement, indicating it is active although the direction differs.
• The vertical force enhances work calculation with its positive y-direction contribution.

Understanding constant force involves recognizing how steady energy transfers impact an object's motion, which is especially practical in calculations like those we encounter here.