A rocket-powered hockey puck is moving on a (frictionless) horizontal air- hockey table. The \(x\) - and \(y\) -components of its velocity as a function of time are presented in the graphs below. Assuming that at \(t=0\) the puck is at \(\left(x_{0}, y_{0}\right)=(1,2)\) draw a detailed graph of the trajectory \(y(x)\).

Understanding velocity components is crucial when dealing with motion in two dimensions. In physics, any velocity vector can be split into two perpendicular components, usually along the x and y axes. For a rocket-powered hockey puck skimming over a frictionless surface, the velocity components, denoted as v_x(t) and v_y(t), vary with time and independently describe the puck's motion along the corresponding axes.

Knowing these components allows us to analyze motion in each direction using the principles of one-dimensional kinematics. Even as they change over time, each component can be represented graphically with a velocity vs. time graph. The true path of the object can be predicted by understanding how these components interact, which is exactly what is required when we want to determine the puck's trajectory on an air-hockey table.

Integration is a mathematical tool that allows us to deduce an object's position from its velocity. The process involves calculating the area under the velocity-time graph for each component, which provides the displacement from the starting position for that component. In the context of the hockey puck's motion, x(t) and y(t)—the position functions—arise from integrating the velocity functions v_x(t) and v_y(t), respectively, with respect to time.

By using the initial conditions of the puck's position, we can formulate explicit equations that describe exactly where the puck will be at any time t. The integral of velocity not only reveals the distance travelled but also incorporates the direction of motion, making it a versatile technique for motion analysis.

Trajectory plotting is the process of determining the path an object will follow through space over time. This concept is widely used in physics to predict the future position of objects in motion. When it comes to our hockey puck, we are interested in demonstrating its trajectory across the air-hockey table, which will be a two-dimensional diagram, showing the path plotted with x as the horizontal and y as the vertical component.

Once the position functions are known, trajectory plotting involves eliminating the time variable to express y directly as a function of x, denoted as y(x). Doing so requires algebraic manipulation and sometimes substitution of one function into another. The resulting graph not only serves as a visual representation but also helps in understanding the dynamics of the puck's movement.

Piecewise functions are mathematical expressions defined by multiple sub-functions, each applying to a certain interval of the independent variable, such as time in our hockey puck example. They're particularly useful when describing phenomena that behave differently during separate phases. For instance, the velocity of the hockey puck might change at distinct moments of time in a stepwise fashion, represented by various segments on the velocity-time graphs.

When solving for the puck's trajectory, we effectively work out piecewise-defined functions for velocity to integrate them and find the puck's position over time. Each segment's integration must be handled separately due to the unique behavior in that interval. Consequently, understanding piecewise functions is essential for accurately modeling systems with changing dynamics, which encapsulates our rocket-powered hockey puck's intriguing journey.