You are a general in the Napoleonic war. The battle has taken you to a trench 7m deep. You have a mortar that will fire an explosive at a speed of 200 m/s at an angle of 80. You have a low flying reconnaissance airplane scouting out the enemy troops at a height of 500 m. Your pilot does not want to fly in the path of your bomb. What distance from your camp would your pilot be well advised to avoid?

Question

You are a general in the Napoleonic war. The battle has taken you to a trench 7m deep. You have a mortar that will fire an explosive at a speed of 200 m/s at an angle of 80. You have a low flying reconnaissance airplane scouting out the enemy troops at a height of 500 m. Your pilot does not want to fly in the path of your bomb. What distance from your camp would your pilot be well advised to avoid?

Answer 1

To determine the distance that the pilot should avoid flying, we need to calculate the range of the mortar shell.

The range of a projectile can be calculated using the horizontal component of its initial velocity and the time of flight. The horizontal component of the velocity can be found using the formula: Vx = V * cos(θ), where V is the initial velocity and θ is the launch angle.

Given:
- Initial velocity, V = 200 m/s
- Launch angle, θ = 80 degrees

First, let's calculate the horizontal component of the velocity:
Vx = 200 * cos(80)
Vx ≈ 34.982 m/s

Next, we need to calculate the time of flight for the mortar shell. Since the shell is fired horizontally, there is no initial vertical velocity. Therefore, we can use the time it takes for the shell to reach its maximum height, which is at the halfway point of its flight, and double it to find the total time of flight.

The formula to find the time of flight to reach maximum height is: t = V * sin(θ) / g, where g is the acceleration due to gravity (approximately 9.8 m/s²).

Using the launch angle θ = 80 degrees:
t = (200 * sin(80)) / 9.8
t ≈ 26.289 s

Therefore, the total time of flight is approximately 2 * 26.289 = 52.578 s.

Now, we can calculate the range by multiplying the horizontal component of the velocity by the time of flight:
Range = Vx * t
Range ≈ 34.982 * 52.578
Range ≈ 1838.705 m

So, the range of the mortar shell is approximately 1838.705 meters.

To determine the distance that the pilot should avoid, we need to add the height of the reconnaissance airplane to the range of the mortar shell: 1838.705 + 500 = 2338.705 meters.

Therefore, the pilot should be well advised to avoid flying within approximately 2338.705 meters from the camp.