What is the cannon's velocity?
You are about to begin a battle during the medieval times. Your enemy has the high ground (about 100 m). You find that the general is at the front lines at the top of the hill. The hill height actually follows the following function:
height = (x-position)^2 (for 0 m<x<10 m)
100 m (x>10m)
What should the total initial velocity of the canon be if you want your first blow to be a direct hit on the general?
Note: ignore air resistance
Should I start with energy conservation laws or kinematics equations?
Answer
As written there seems to be two interpretations of this problem: a traditional cannon that will need to clear the hill and a cannon shooting the cannonball along the ground to fly off the hill. Since the first interpretation would require the angle of the cannon, which is not given, and would not need an exact function for the hill, I will assume the second interpretation. If the first interpretation is correct, this should just be solely a kinematics problem with a minimum angle necessary.
This problem is best solved by working backwards. With some basic calculus, you can determine the angle the cannonball will be launched at when it flies off the hill. Using this angle and the distance from this point to the general (it won't be 100m), you should be able to find the velocity need to hit the general directly. Since this point is at the same height as the general, you can simply use the Range equation. Now that you have a velocity at the top of the hill's slope, you can use conservation of energy to determine the velocity at the bottom of the slope, which should be your initial velocity.
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As written there seems to be two interpretations of this problem: a traditional cannon that will need to clear the hill and a cannon shooting the cannonball along the ground to fly off the hill. Since the first interpretation would require the angle of the cannon, which is not given, and would not need an exact function for the hill, I will assume the second interpretation. If the first interpretation is correct, this should just be solely a kinematics problem with a minimum angle necessary.
This problem is best solved by working backwards. With some basic calculus, you can determine the angle the cannonball will be launched at when it flies off the hill. Using this angle and the distance from this point to the general (it won't be 100m), you should be able to find the velocity need to hit the general directly. Since this point is at the same height as the general, you can simply use the Range equation. Now that you have a velocity at the top of the hill's slope, you can use conservation of energy to determine the velocity at the bottom of the slope, which should be your initial velocity.