How long (in s) would the ball have been in the air before it bounced if the ball was hit at a height of 1.1 m? How far from the batter (horizontal distance) would the ball have tr... How long (in s) would the ball have been in the air before it bounced if the ball was hit at a height of 1.1 m? How far from the batter (horizontal distance) would the ball have travelled (in m) before it hit the ground?
Understand the Problem
The question is asking for calculations related to projectile motion, specifically how long a ball would remain in the air when hit horizontally from a height of 1.1 meters, as well as how far it would travel horizontally before hitting the ground.
Answer
The ball would have been in the air for approximately $0.47 \, \text{s}$ and would travel a horizontal distance of approximately $12.69 \, \text{m}$.
Answer for screen readers
The time the ball would have been in the air is approximately $0.47 , \text{s}$, and the horizontal distance it would have travelled is approximately $12.69 , \text{m}$.
Steps to Solve
- Calculate time in the air To find the time the ball remains in the air, we can use the formula for free fall due to gravity: $$ d = \frac{1}{2} g t^2 $$ where:
- ( d = 1.1 , \text{m} ) (the height),
- ( g = 9.81 , \text{m/s}^2 ) (acceleration due to gravity).
Rearranging the formula for ( t ): $$ t = \sqrt{\frac{2d}{g}} = \sqrt{\frac{2 \times 1.1}{9.81}} $$
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Substituting the values Calculating the value: $$ t = \sqrt{\frac{2 \times 1.1}{9.81}} \approx \sqrt{0.224} \approx 0.47 , \text{s} $$
-
Calculate horizontal distance To find the horizontal distance ( x ), use the formula: $$ x = v t $$ where:
- ( v = 27 , \text{m/s} ) (horizontal velocity),
- ( t ) is the time found previously.
Substituting the values: $$ x = 27 \times 0.47 $$
- Final Calculation for horizontal distance Calculating ( x ): $$ x \approx 12.69 , \text{m} $$
The time the ball would have been in the air is approximately $0.47 , \text{s}$, and the horizontal distance it would have travelled is approximately $12.69 , \text{m}$.
More Information
These calculations show how projectile motion operates, separating the vertical and horizontal components. The time of flight depends solely on the vertical drop, while the horizontal distance is determined by the horizontal speed and the time in the air.
Tips
- Forgetting to convert units (if needed).
- Misapplying the formula for time in free fall—ensure using the correct values for distance and gravity.
- Not distinguishing between vertical and horizontal motion in projectile problems.
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