The illustration for this problem can be seen at screenshots(dot)firefox(dot)com/atwXACAA9akaoAji/null

The army is testing out a new prototype artillery cannon with an uncommonly high muzzle velocity of 1000 m/sec . The design bugs haven't been fully worked out yet, so the cannon has to wait at least a full minute between shots (reloading, etc.). The cannon can be aimed with an angle between 0 and 90 degrees with respect to the horizontal.

A particularly demanding exercise is being conducted, in which the cannon fires an initial shot into the air, and while the first shell is still in the air, a second shot is fired (at a different angle) so that both shells impact the target (at ground level) simultaneously.

What is the maximum horizontal distance from the cannon to the target, such that this is possible?

Details and Assumptions:

Assume level ground, with no air resistance.
The gravitational acceleration is 10 m/s².
Give your answer in meters, to the nearest whole meter.
For the sake of this problem, ignore the Earth's curvature.

so, how far have you gotten? We know that the horizontal distance at time t is

d = v cosθ t

So, for this problem, if we let
θ = angle of 1st shot
Ø = angle of 2nd shot
k = delay between shots
t = flight time of 1st shot

1000 cosθ t = 1000 cosØ (t-k)
cosθ/cosØ = (t-k)/t

and we know that the vertical speed of the 1st shot is v(t) = 1000 sinθ - 10t
so that it spends t=200sinθ seconds in the air.

So, now we have
cosθ/cosØ = 1 - k/(200sinθ)

So, you need to find the max value of t (or sinθ, since t=200sinθ) which satisfies these conditions.

To solve this problem, we need to analyze the motion of the projectile fired from the cannon. The key is to find the conditions under which both projectiles, fired at different angles, will hit the ground simultaneously.

Let's break down the problem step by step:

Step 1: Calculate the time of flight for the first projectile:
The time of flight for a projectile can be calculated using the formula:
time = (2 * velocity * sin(angle)) / acceleration

Given:
- Velocity of the projectile = 1000 m/s
- Angle of projection for the first projectile = unknown (let's call it angle1)
- Acceleration due to gravity = 10 m/s²

Using the formula, we can calculate the time it takes for the first projectile to reach the ground.

Step 2: Calculate the maximum height reached by the first projectile:
The maximum height reached by a projectile can be calculated using the formula:
max_height = (velocity^2 * sin^2(angle)) / (2 * acceleration)

Using the known values, we can calculate the maximum height reached by the first projectile.

Step 3: Calculate the time of flight for the second projectile:
Since we want both projectiles to reach the ground simultaneously, the time of flight for the second projectile should be the same as that for the first projectile (calculated in Step 1).

Step 4: Calculate the horizontal distance traveled by the second projectile:
The horizontal distance traveled by a projectile can be calculated using the formula:
horizontal_distance = velocity * cos(angle) * time

Given:
- Velocity of the projectile = 1000 m/s
- Angle of projection for the second projectile = unknown (let's call it angle2)
- Time of flight for the second projectile (from Step 3)

Using the formula, we can calculate the horizontal distance traveled by the second projectile.

Step 5: Find the maximum horizontal distance:
We want to find the maximum horizontal distance from the cannon to the target, such that both projectiles hit the ground simultaneously. This implies that the horizontal distances traveled by both projectiles should be the same.

So, we need to find the maximum horizontal distance where:
horizontal_distance1 = horizontal_distance2

Now we have two equations with two unknowns (angle1 and angle2). We can solve these equations to find the values of angle1 and angle2, and then calculate the maximum horizontal distance from the cannon to the target.

To solve the equations, you can use numerical methods, graphing calculators, or programming languages like Python to find the values of angles and the maximum horizontal distance.

Note: The screenshot provided in the question is not accessible, so I am unable to refer to it directly. However, by following the steps outlined above, you should be able to find the solution to the problem.