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?

Question

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?

Answer 1

To determine the angle at which you should tilt your cannon and the firing time to hit the enemy just as they reach the edge of the forest, we can apply basic principles of projectile motion.

First, let's find the time at which the enemy reaches the edge of the forest. We can use the equation:

distance = speed * time

Given that the enemy cavalry charge is 1000m away from the edge of the forest and moving at a speed of 7m/s, we can calculate the time it takes for them to reach the forest:

time = distance / speed
time = 1000m / 7m/s
time ≈ 142.86s

Therefore, it will take approximately 142.86 seconds for the enemy to reach the edge of the forest.

Next, we need to determine the horizontal distance covered by the cannonball during this time. The horizontal distance can be calculated using the equation:

horizontal distance = horizontal velocity * time

In this case, the horizontal velocity is the component of the initial velocity of the cannonball in the horizontal direction. Given that the cannonball is shot at a speed of 350 m/s, the horizontal velocity can be calculated using trigonometry:

horizontal velocity = initial velocity * cos(angle)

Now, let's solve for the angle. We know that the horizontal distance should be equal to 800m (the distance to the forest) when the enemy cavalry charge reaches the forest.

800m = horizontal velocity * time
800m = (initial velocity * cos(angle)) * time

Substituting the known values:

800m = (350m/s * cos(angle)) * 142.86s

Simplifying the equation, we can solve for the angle:

cos(angle) = (800m / 350m/s * 142.86s)
angle = arccos(800m / (350m/s * 142.86s))

Using a scientific calculator, we can find that the angle is approximately 44.86 degrees.

So, you must tilt your cannon at an angle of approximately 44.86 degrees, and you should fire it 142.86 seconds before the enemy reaches the edge of the forest to hit them just as they arrive.

Please note that in reality, there are other factors to consider, such as air resistance, elevation changes, and the specific characteristics of the cannonball, which could affect the accuracy of the shot.