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 HH)

  • The cannonball is launched horizontally, so the initial vertical velocity (vy0v_{y0}) is 00.

  • Vertical motion is influenced by gravity. The time (tt) it takes for the cannonball to fall from the height HH is given by the equation: H=12gt2H = \frac{1}{2} g t^2 Solving for tt: t=2Hgt = \sqrt{\frac{2H}{g}}

  • Horizontal motion is constant since there's no horizontal acceleration. If vxv_x is the horizontal velocity, the horizontal distance (DD) covered is: D=vxtD = v_x t Substituting tt: D=vx2HgD = v_x \sqrt{\frac{2H}{g}}

2. Second Launch (Cliff Height 2H2H)

  • Using the same logic, for the second launch, the time (tt') it takes to fall from the height 2H2H is: 2H=12gt22H = \frac{1}{2} g t'^2 Solving for tt': t=4Hg=2Hgt' = \sqrt{\frac{4H}{g}} = 2\sqrt{\frac{H}{g}}

  • The horizontal distance (DD') covered in this case, with the same horizontal velocity vxv_x, is: D=vxtD' = v_x t' Substituting tt': D=vx×2HgD' = v_x \times 2\sqrt{\frac{H}{g}}

3. Relationship Between Distances

From the first launch, we have D=vx2HgD = v_x \sqrt{\frac{2H}{g}}. We can express vxv_x in terms of DD and HH: vx=D2Hgv_x = \frac{D}{\sqrt{\frac{2H}{g}}}

Now substitute vxv_x into DD': D=D2Hg×2HgD' = \frac{D}{\sqrt{\frac{2H}{g}}} \times 2\sqrt{\frac{H}{g}}

Simplifying this: D=D×2D' = D \times \sqrt{2}

Conclusion

When the cannonball is launched from a cliff of height 2H2H, it will land a horizontal distance of D×2D \times \sqrt{2} from the base of the cliff. If the original distance DD is given, then the new distance is approximately 1.41D1.41D.

Would you like further clarification or more details on this?

Here are some related questions you might find interesting:

  1. How does changing the launch angle affect the horizontal distance?
  2. What if the cannonball was launched at an angle instead of horizontally?
  3. How would air resistance impact the motion of the cannonball?
  4. What is the maximum height achieved by a projectile launched at an angle?
  5. 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