Suppose you are a general in the Napoleonic wars. You are no top of a plain 50 m high overlooking the enemy soldiers. You now have a brand new cannon that will decimate the enemy. Unfortunately in the christening process you used some rather cheap grade champagne and all the water in the champagne rusted the hinges of the cannon so that it is stuck angled upward at an angle of 30 from the horizontal. Your trusty new cannon expert insisted that the ball will reach a maximum height of 150m from the ground but unfortunately he was killed before he could tell you where the cannonball would land on the plain below. You have soldiers on the ground and you want to make sure that only the enemy soldiers are killed by your uber cannon. Determine the range of the cannon in order to keep your soldiers safe.

Question

Suppose you are a general in the Napoleonic wars. You are no top of a plain 50 m high overlooking the enemy soldiers. You now have a brand new cannon that will decimate the enemy. Unfortunately in the christening process you used some rather cheap grade champagne and all the water in the champagne rusted the hinges of the cannon so that it is stuck angled upward at an angle of 30 from the horizontal. Your trusty new cannon expert insisted that the ball will reach a maximum height of 150m from the ground but unfortunately he was killed before he could tell you where the cannonball would land on the plain below. You have soldiers on the ground and you want to make sure that only the enemy soldiers are killed by your uber cannon. Determine the range of the cannon in order to keep your soldiers safe.

Answer 1

See Related Questions: Wed,2-19-14,9:27

PM.

Answer 2

To determine the range of the cannon in order to keep your soldiers safe, we need to find out where the cannonball will land on the plain below. Given that the cannon is stuck angled upward at an angle of 30° from the horizontal, and the maximum height of the cannonball is 150m, we can use some basic physics equations to solve this problem.

First, let's break down the motion of the cannonball into horizontal and vertical components.

1. Vertical Motion:
The vertical motion of the cannonball can be described by the equation:
h = (v₀² * sin²θ) / (2 * g)
where h is the maximum height (150m), v₀ is the initial velocity of the cannonball, θ is the launch angle (30°), and g is the acceleration due to gravity (9.8 m/s²).

Rearranging the equation, we can solve for v₀:
v₀ = √((h * 2 * g) / sin²θ)
v₀ = √((150 * 2 * 9.8) / sin²30°)
v₀ ≈ 313.2 m/s

2. Horizontal Motion:
The horizontal motion of the cannonball can be described by the equation:
R = (v₀² * sin2θ) / g
where R is the range of the cannonball.

Rearranging the equation, we can solve for R:
R = (v₀² * sin60°) / g
R = (313.2² * √3/2) / 9.8
R ≈ 5439.9 m

Therefore, the range of the cannon is approximately 5439.9 meters. To keep your soldiers safe, ensure that they are positioned beyond this range from the cannon.