A bomber is moving horizontally with 195 km/h. It intends to aim at a factory 550 m below. At what horizontal distance from the target should it release the bomb?
Understand the Problem
The question is asking us to calculate the horizontal distance from the target where the bomb should be released, given the speed of the bomber and the vertical distance to the target.
Answer
$$ d = v \cdot \sqrt{\frac{2h}{g}} $$
Answer for screen readers
The horizontal distance ( d ) from the target where the bomb should be released is given by: $$ d = v \cdot \sqrt{\frac{2h}{g}} $$
Steps to Solve
- Identify Variables Let:
- ( d ) be the horizontal distance from the target where the bomb should be released.
- ( v ) be the speed of the bomber (in units like meters per second).
- ( h ) be the vertical distance to the target (in the same units like meters).
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Determine Time to Fall We know that the time ( t ) it takes for the bomb to fall from the height ( h ) can be calculated using the formula for free fall: $$ h = \frac{1}{2} g t^2 $$ Where ( g ) is the acceleration due to gravity (approximately ( 9.81 , \text{m/s}^2 )). Rearranging the formula gives: $$ t = \sqrt{\frac{2h}{g}} $$
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Calculate Horizontal Distance The horizontal distance ( d ) can be found using the speed of the bomber and the time it takes for the bomb to hit the target: $$ d = v \cdot t $$ Substituting the expression for ( t ) from the previous step, we get: $$ d = v \cdot \sqrt{\frac{2h}{g}} $$
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Plug in Known Values Insert the values of ( v ) (the bomber speed) and ( h ) (the vertical distance) into the equation for ( d ) to calculate the exact horizontal distance.
The horizontal distance ( d ) from the target where the bomb should be released is given by: $$ d = v \cdot \sqrt{\frac{2h}{g}} $$
More Information
This problem involves concepts of projectile motion and can be very useful in fields such as physics and engineering. Knowing how to calculate the right distance ensures accuracy in bomb release, which could be critical in various applications.
Tips
- Forgetting to convert units: Ensure that the speed and distance are in compatible units (e.g., meters and seconds).
- Miscalculating the time of fall: Double-check the rearrangement of the free fall formula to avoid errors in the time calculation.
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