Math Problem Statement
A cannon launches a cannonball horizontally from a cliff of height H lands a horizontal distance D from the base of the cliff. If the same cannon launches an identical cannonball horizontally from a cliff of height 2H, how far will it land?
Solution
Let's break down the problem step-by-step using the principles of projectile motion.
1. First Launch (Cliff Height )
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The cannonball is launched horizontally, so the initial vertical velocity () is .
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Vertical motion is influenced by gravity. The time () it takes for the cannonball to fall from the height is given by the equation: Solving for :
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Horizontal motion is constant since there's no horizontal acceleration. If is the horizontal velocity, the horizontal distance () covered is: Substituting :
2. Second Launch (Cliff Height )
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Using the same logic, for the second launch, the time () it takes to fall from the height is: Solving for :
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The horizontal distance () covered in this case, with the same horizontal velocity , is: Substituting :
3. Relationship Between Distances
From the first launch, we have . We can express in terms of and :
Now substitute into :
Simplifying this:
Conclusion
When the cannonball is launched from a cliff of height , it will land a horizontal distance of from the base of the cliff. If the original distance is given, then the new distance is approximately .
Would you like further clarification or more details on this?
Here are some related questions you might find interesting:
- How does changing the launch angle affect the horizontal distance?
- What if the cannonball was launched at an angle instead of horizontally?
- How would air resistance impact the motion of the cannonball?
- What is the maximum height achieved by a projectile launched at an angle?
- How can you derive the time of flight for a projectile launched at an arbitrary angle?
Tip: In projectile motion, horizontal and vertical motions are independent of each other. Horizontal motion has constant velocity, while vertical motion is influenced by gravity.
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Math Problem Analysis
Mathematical Concepts
Projectile Motion
Kinematics
Horizontal and Vertical Motion Independence
Formulas
Time of flight: t = sqrt(2H/g)
Horizontal distance: D = v_x * t
t' = sqrt(4H/g)
D' = v_x * t'
Theorems
Independence of horizontal and vertical motion in projectile motion
Relationship between time of flight and vertical distance
Suitable Grade Level
Grades 10-12
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