Question

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 $(x_0,y_0)=(1,2)$ draw a detailed graph of the trajectory $y(x)$.

Short answer

Question: Find the trajectory of a hockey puck given its initial position (1, 2) and the graphs of its velocity components as functions of time. Answer: To find the trajectory y(x) of the hockey puck, follow these steps: 1. Analyze the given graphs of the velocity components and obtain their functional forms. 2. Integrate the velocity component functions with respect to time and add the initial positions to obtain the position functions x(t) and y(t). 3. Eliminate the time parameter from the position functions by solving for t in x(t) and substituting it into y(t). 4. Simplify the equation to get the final expression for y(x). 5. Plot the trajectory y(x) on a coordinate plane. Note: The actual trajectory equation y(x) and its graph depend on the specific velocity functions provided in the problem.

Step-by-step solution

Find position components as functions of time

To find the position components as functions of time, we need to integrate the velocity components with respect to time. Let's denote the $x$-component of the velocity as $v_x(t)$ and the $y$-component as $v_y(t)$. We are given their respective graphs. From those graphs, we can find the respective velocity functions and derive the position functions as follows: 1. Integrate $v_x(t)$ with respect to $t$ and add the initial $x$-position, i.e., $x(t)=\int v_x(t)dt+x_0$. 2. Integrate $v_y(t)$ with respect to $t$ and add the initial $y$-position, i.e., $y(t)=\int v_y(t)dt+y_0$.

Obtain the velocity functions from the given graphs

From the given graphs of $v_x(t)$ and $v_y(t)$, obtain the functional forms of the velocity components. They are likely piecewise functions, reflecting different parts of the graphs. For example, if $v_x(t)$ is a linear function with a negative slope from $0$ to $t_1$, and then an increasing linear function with positive slope from $t_1$ to $t_2$, then it can be represented as a piecewise function such as: $v_x(t)= $$\{\begin{array}{ll}at+b,& \text{for }0\le t\le t_1\\ ct-d,& \text{for }{t}_1<t\le t_2\end{array}$$ $ Perform a similar analysis for the $v_y(t)$ graph to obtain the functional form of the $y$-velocity component.

Integrate the velocity functions and find the position functions

Once the functional forms of the velocity components are obtained, integrate them with respect to time to get the position functions using the initial positions $x_0=1$ and $y_0=2$: 1. $x(t)=\int v_x(t)dt+x_0$. Calculate this integral while considering the piecewise function. 2. $y(t)=\int v_y(t)dt+y_0$. Calculate this integral while considering the piecewise function as well.

Eliminate the time parameter to obtain y(x)

Now we have the position components as functions of time, i.e., $x(t)$ and $y(t)$. To find the trajectory $y(x)$, we need to eliminate the time parameter $t$. 1. Solve the $x(t)$ equation for $t$ symbolically. 2. Substitute the resulting expression for $t$ in the $y(t)$ expression. 3. Simplify the equation obtained after substitution to get the final expression for $y(x)$.

Plot the trajectory y(x)

Using the equation for $y(x)$ obtained in the previous step, draw the graph of the trajectory on the coordinate plane. Make sure to label axes, indicate the starting point, and indicate the direction of motion along the path. Now, you have the requested detailed graph of the trajectory $y(x)$.

Key concepts

Velocity Components

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 of Velocity

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

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

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.