A rocket-powered hockey puck is moving on a (friction-less) 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 (X0,Y0)=(1,2),
draw a detailed graph of the trajectory y(x).
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To draw a detailed graph of the trajectory y(x) for the rocket-powered hockey puck, we need to analyze the x- and y-components of its velocity as a function of time.
Looking at the graphs presented, we can see that the x-velocity of the puck remains constant at 1 m/s over time. This means that the puck is moving horizontally in the positive x-direction at a constant speed.
On the other hand, the y-velocity of the puck changes over time. From t = 0 to t = 2 s, the y-velocity increases linearly from 2 m/s to 6 m/s. Then, from t = 2 s to t = 4 s, the y-velocity decreases linearly back to 2 m/s.
To determine the y-position of the puck as a function of time, we need to integrate the y-velocity with respect to time. However, since we do not have an equation for the y-velocity as a function of time, we can approximate it by connecting the data points in the graph.
Based on the given information, we can draw the following trajectory graph:
1) Start at the initial position (X0, Y0) = (1, 2).
2) The x-coordinate of the puck remains constant at 1 m/s, so the x-axis is a straight horizontal line.
3) From t = 0 to t = 2 s, the y-velocity increases linearly from 2 m/s to 6 m/s. This means that the slope of the y(x) curve in this region is positive, indicating an upward trajectory.
4) From t = 2 s to t = 4 s, the y-velocity decreases linearly back to 2 m/s. This means that the slope of the y(x) curve in this region is negative, indicating a downward trajectory.
Connecting the corresponding points on the graph with smooth curves will give you the trajectory y(x) of the rocket-powered hockey puck.
Note: Without knowing the exact shape of the y-velocity curve, we can only draw an approximate trajectory graph based on the available information.