91Ó°ÊÓ

When dealing with motion problems, particularly those in a two-dimensional plane, breaking the overall velocity into its components is essential. Imagine velocity as a vector, which means it has both a magnitude and a direction. For a ball rolling in the fourth quadrant at an angle, we can improve accuracy by focusing on velocity components. This effectively simplifies the problem into manageable parts.
The effective approach is to determine the x and y components of the velocity. These components express how much of the ball's velocity is acting along the horizontal (x-axis) and vertical (y-axis) directions.

• In our problem, the angle given is 45°, crucial for calculating components because at 45°, the values of sine and cosine are equal. This symmetry allows each component to have equal magnitude.
• Using trigonometric functions, specifically cosine for the x-component and sine for the y-component, we find each: \[ v_x = v \, \cos(45°) = v_y = v \, \sin(45°). \]
• For an overall velocity of 1.50 m/s, the components become about 1.061 m/s for both x and y directions, adjusted for their respective signs.

Recognizing these components' direction in a quadrant is key, ensuring the x is positive and y is negative since we're in the fourth quadrant.

Now that we know how to break down the velocity, calculating displacement becomes straightforward. Displacement simply refers to how far an object has moved from its initial position over a given time frame. For these problems, knowing the velocity components lets us quickly find out how much distance was covered in each direction.
To find displacement, multiply the velocity component by time:

• The displacement in the x-direction, given by \( x = v_x \times t \), results in: \[ x = 1.061 \, \text{m/s} \times 1.65 \, \text{s} \approx 1.751 \, \text{meters}. \]
• Similarly, for the y-direction using \( y = v_y \times t \), we have: \[ y = -1.061 \, \text{m/s} \times 1.65 \, \text{s} \approx -1.751 \, \text{meters}. \]

Note the negative sign for the y-direction due to moving downward below the x-axis. Using these displacements, always remember they are changes from an initial position, usually from the origin in these setups.

Coordinate systems form the backbone of analyzing motion in physics, especially in two dimensions. They allow us to visually and mathematically represent positions and movements, breaking down complex trajectories into understandable components.
In this exercise, the ball starts at the origin at time zero, with coordinates (0,0). The coordinate plane is split into four quadrants:

• First quadrant: both x and y are positive.
• Fourth quadrant, where our ball moves: x is positive and y is negative.

This structure helps determine the direction of each velocity component. By knowing which direction (positive or negative) each axis moves towards, you can accurately plot the path of an object.
After calculating displacement, finding the coordinates merely involves adjusting from the initial origin location. The new position, ultimately, reflects the total extent of distance traveled in each direction. This method is fundamental in physics problems, providing a clear path from initial standing to a conclusive answer, with the ball at approximately (1.751, -1.751). Remember, coordinate systems are all about understanding the direction and magnitude of every move on a grid.