Bob and Alice both throw baseballs vertically upwards from the same starting height. Both balls take the same amount of time to go up to their maximum height. Which of the follow is true (select all that apply)

Question

Bob and Alice both throw baseballs vertically upwards from the same starting height. Both balls take the same amount of time to go up to their maximum height.  Which of the follow is true (select all that apply)

BlackTom AI · Accepted Answer

We start by identifying what is given and what must be true for vertical motion under gravity with no air resistance.

- The two balls are thrown upward from the same starting height.
- They both take the same amount of time to reach their maximum height.
- In vertical motion with only gravity, the time to reach maximum height depends only on the initial vertical speed, via t_up = v0/g, where v0 is the initial upward speed.

Now, evaluate each option in turn:

Option 1: 'Both Bob and Alice must have thrown the balls with the same speed'
- Since t_up = v0/g, equal times to reach max height imply equal v0 for both balls. Therefore, this statement is consistent with the given information and true.

Option 2: 'Bob could have thrown his ball with a larger speed than Alice'
- If Bob’s initial speed were larger, his t_up would be larger (t_up = v0/g). Since the observed times are equal, a larger initial speed for Bob cannot be correct given the same t_up. This option contradicts the given condition and is false.

Option 3: 'Alice could have thrown her ball with a larger speed than Bob'
- The same reasoning as for option 2 applies in reverse: a larger speed for Alice would imply a longer t_up, not equal times. Therefore this option is false.

Option 4: 'Both balls must have reached the same maximum height'
- The maximum height is h = v0^2/(2g). Since we established that v0 is the same for both balls (from equal t_up), the maximum heights must be the same as well. Thus this statement is true.

Option 5: 'Alice's ball could have reached a large maximum height compared to Bob's'
- A larger maximum height would require a larger initial speed for Alice, which would also lengthen t_up beyond Bob’s, contradicting the given equal times. This option is false.

Option 6: 'Bob's ball could have reached a large maximum height compared to Alice's'
- For the same reason as option 5, a larger maximum height for Bob would require a greater initial speed, which would change the time to reach max height. This option is false.

In summary, the correct conclusions supported by the physics are that both balls were launched with the same initial speed and both reached the same maximum height, while all statements suggesting unequal speeds or unequal maximum heights are inconsistent with t_up being equal.

Bob and Alice both throw baseballs vertically upwards from the same starting height. Both balls take the same amount of time to go up to their maximum height. Which of the follow is true (select all that apply) 多项选择题

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