You are general in the Napoleonic wars. You are on top of a plain 12m talk and you have a cannon that can shoot a cannonball out at 350 m/s. 800 m away is a thick forest that you cannot see into but one of your scouts in a hot air balloon has just signaled to you that an enemy cavalry charge inside the forest is rushing toward you at a speed of 7m/s and they are 1000m from the edge of the forest. You want their entry onto the battle plain to welcomed with a nice big fat cannonball. At what angle must you tilt your cannon so the cannonball hits the edge of the forest and what time (starting with t=0 sec at the present) must you fire it so that it his the enemy just as they reach the forests edge?

To calculate the angle at which you must tilt your cannon and the time at which you should fire it to hit the enemy cavalry at the moment they reach the edge of the forest, we can break down the problem into horizontal and vertical components.

First, let's calculate the time it takes for the enemy cavalry to reach the edge of the forest. Using the equation d = vt, where d is the distance, v is the velocity, and t is the time, we can plug in the values:

Distance = 1000 m
Velocity = 7 m/s

t = d / v
t = 1000 m / 7 m/s
t ≈ 142.85 s

Now, let's determine the horizontal distance the cannonball needs to travel to hit the edge of the forest. The enemy cavalry will be moving in the horizontal direction for the same amount of time as calculated above, but at a different speed. Using the equation d = vt again:

Velocity = 7 m/s
t = 142.85 s

Distance = Velocity * time = 7 m/s * 142.85 s
Distance ≈ 1000 m

This means that if the cannonball is shot at a horizontal distance of 1000 m away from you, it will hit the edge of the forest at the same time as the enemy cavalry.

To determine the required angle of the cannon, we can use the formula for the horizontal and vertical components of projectile motion:

Horizontal distance = velocity * time * cos(angle)
Vertical distance = velocity * time * sin(angle) - (0.5 * g * t^2)

We can rearrange the horizontal distance equation to solve for the angle:

angle = arccos(horizontal distance / (velocity * time))

angle = arccos(1000 m / (350 m/s * 142.85 s))

Calculating this value will give you the angle at which you need to tilt your cannon. However, please note that due to potential variations and uncertainties in the calculations, it is vital to double-check and adjust as necessary before firing the cannon. Additionally, factors such as wind speed and direction may impact the accuracy of the shot.