Question

lesson 8: interpreting and creating graphs
cool down: caught in a tree
a child tosses a baseball up into the air. on its way down, it gets caught in a tree for several seconds before falling down to the ground.
sketch a graph that represents the height of the ball, h, as a function of time, t.
be sure to include a label and a scale for each axis.

Explanation:

Step1: Analyze the motion phases

The baseball's motion has three phases: rising, being caught in the tree (constant height), and falling. When tossed up, height \( h \) increases with time \( t \) until it reaches a peak. Then, as it falls, height decreases until it gets caught in the tree (height becomes constant for a few seconds). After that, it falls to the ground (height decreases to 0).

Step2: Sketch the graph structure

• Axes Labeling: Label the horizontal axis as "Time (\( t \)) (seconds)" and the vertical axis as "Height (\( h \)) (meters, or any unit for height)".
• Scale: Choose a reasonable scale, e.g., each unit on the time axis represents 1 second, and each unit on the height axis represents 1 meter (adjust based on expected maximum height).
• Graph Shape:
• Start at \( (0, h_0) \) where \( h_0 \) is the initial height (e.g., 1 meter if the child is holding it at that height).
• Draw a curve (or line, for simplicity, assuming constant acceleration) increasing to a peak (e.g., at \( t = 2 \) seconds, \( h = 5 \) meters).
• Then draw a decreasing line until it reaches a height where it gets caught (e.g., at \( t = 3 \) seconds, \( h = 3 \) meters), then draw a horizontal line (constant height) for a few seconds (e.g., from \( t = 3 \) to \( t = 5 \) seconds), then a decreasing line to \( (t, 0) \) (e.g., at \( t = 7 \) seconds, \( h = 0 \)).

Answer:

The graph should have:

• Horizontal axis: Time (\( t \)) with scale (e.g., 1 unit = 1 second).
• Vertical axis: Height (\( h \)) with scale (e.g., 1 unit = 1 meter).
• Shape: Increasing curve (or line) to a peak, decreasing to a constant height (horizontal line) for a few seconds, then decreasing to 0. (A hand - drawn sketch following this description is the solution, with axes labeled and scaled appropriately.)