You're sailboarding at \(6.5 \mathrm{m} / \mathrm{s}\) when a wind gust hits, lasting \(6.3 \mathrm{s}\) accelerating your board at \(0.48 \mathrm{m} / \mathrm{s}^{2}\) at \(35^{\circ}\) to your original direction. Find the magnitude and direction of your displacement during the gust.

Displacement represents how far out of place an object is; it's the object's overall change in position. In physics, it's crucial to distinguish between distance and displacement. Distance refers to the total ground covered, while displacement measures the shortest path between the starting and ending points and is direction-sensitive.

In the context of the sailboarding problem, displacement is calculated by finding the resultant vector that combines the forward motion and the additional velocity provided by the wind gust. It gives the sailboarder’s change in position from the start to the end of the gust. Displacement, being a vector, must account not only for magnitude but also for the direction of the sailboarder's movement.

Vectors are quantities that have both magnitude and direction. When dealing with motion, it’s often useful to break vectors down into components along mutually perpendicular axes, commonly the x (horizontal) and y (vertical) axes.

For the sailboarding scenario, the initial velocity has a component along the x-axis and none along the y-axis before the gust of wind changes its angle of motion. Similarly, acceleration due to the wind is broken down into x and y components. The x-component affects motion parallel to the original direction, and the y-component affects motion perpendicular to it. By understanding vector components, we can simplify the analysis of motion and use scalar methods to solve vector problems.

Understanding Acceleration

Acceleration is the rate of change of velocity. It could involve an increase or decrease in the speed or a change in direction. In the example of sailboarding, acceleration is caused by the wind gust and is in a different direction to the initial velocity.

Calculating Acceleration Components

The problem requires breaking down the acceleration into its x and y components to independently analyze how the wind gust affects the board’s velocity in each direction. The concept of acceleration is vital for predicting how motion changes over time, which is essential for solving kinematic problems.

Motion in two dimensions, or 2D motion, involves moving in a plane, requiring both x and y coordinates for a complete description. This is often managed by treating the motion independently along each axis.

In the sailboarding example, the board moves horizontally while also being pushed at an angle due to the wind gust. By analyzing the motion separately along the x and y axes using the vector components, we can apply kinematic equations to each direction. After calculating the separate displacements, we recombine them using vector addition to obtain the total displacement. This process showcases how 2D motion problems can be dissected into simpler, one-dimensional problems that are easier to solve.