Physics students drop a basketball from 5 feet above the ground and its height is measured each tenth of a second until it stops bouncing. The height of the basketball, h, is clearly a function of the time, i, since it was dropped. (a) Sketch the general graph of what you believe this function would look like. (b) Is the height of the ball a one-to-one function of time? Explain your answer.

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Physics students drop a basketball from 5 feet above the ground and its height is measured each tenth of a second until it stops bouncing. The height of the basketball, h, is clearly a function of the time, i, since it was dropped. (a) Sketch the general graph of what you believe this function would look like. (b) Is the height of the ball a one-to-one function of time? Explain your answer.

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(a) Sketch the general graph of what you believe this function would look like.Since I cannot provide an actual sketch, I will describe what the graph of the height of a bouncing basketball over time would look like:1. The graph would start at the point (0, S) where S is the initial height from which the ball was dropped.2. The graph would then show a decreasing curve as the ball falls towards the ground due to gravity, reaching a height of 0 at the time of the first bounce.3. Upon hitting the ground, the ball would bounce up, but not to the original height S due to energy loss, so the next peak would be lower.4. The graph would show a series of peaks and troughs, with each peak lower than the previous one, and each trough reaching closer to 0 height, until the ball comes to rest.5. The time between bounces would also decrease as the ball loses energy and the bounces become smaller and more frequent.6. Eventually, the graph would show the ball coming to rest at a height of 0.The graph would resemble a damped oscillation, with the amplitude of the oscillations decreasing over time.(b) Is the height of the ball a one-to-one function of time? Explain your answer.No, the height of the ball is not a one-to-one function of time. A one-to-one function means that for each value of the independent variable (in this case, time t), there is exactly one unique value of the dependent variable (height h). However, during the ball's trajectory, there will be multiple instances in time where the ball passes through the same height while going up and coming down between bounces. This means that for certain heights, there are at least two different times at which the ball will be at that height (once on the way up and once on the way down), violating the one-to-one criterion.For example, if we consider the points given in the picture details, at t=0.1 seconds the ball might be at a height of 4 feet on its way down, and at some later time, it might be at 4 feet again on its way up after a bounce. This would mean that the function h(t) is not one-to-one since the height of 4 feet corresponds to at least two different times.

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