A Cannon/Cliff Physics Question?

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In summary, a cannon is fired straight upward from the bottom of a cliff 85 m high and just reaches the top. The initial speed is calculated to be 40.8375 m/s. In the second part, a cannon is placed at the top of the cliff and fired horizontally with the same initial speed, and it is shown numerically that the maximum range is the same as the range from the base of the cliff. The equation used for the vertical part is not relevant for this question, and the range equation is needed to find the numerical solution.
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popmop2
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Homework Statement


A cannon is placed at the bottom of a cliff 85 m high. If the cannon is fired straight upward, the cannonball just the reaches the top of the cliff.
a) Calculate the initial speed of the cannonball.
b) Suppose a second cannon is placed at the top of the cliff and fired horizontally with the same initial speed as part (a). Prove numerically that the range of this cannon is the same as the maximum range of the cannon from the base of the cliff.


Homework Equations


for a) v^2 = 2gh
for b) ymax = (v^2 sin^2 ∅)/2g


The Attempt at a Solution



for a) i got v = 40.8375 m/s

for b) i got ymax = 85 meters but that is only for the vertical part.

I am not sure how to find the numerically part
 
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  • #2
popmop2 said:
for b) ymax = (v^2 sin^2 ∅)/2g
Not a useful equation for this question. Do you have any equations relating to range?
 

1. How does the height of a cliff affect the trajectory of a cannonball?

The height of a cliff can significantly affect the trajectory of a cannonball. The higher the cliff, the longer the cannonball will stay in the air and the farther it will travel. This is because the cannonball has more time to fall and be affected by gravity, resulting in a longer horizontal distance traveled.

2. What is the relationship between the angle of the cannon and the distance the cannonball travels?

The angle of the cannon is directly related to the distance the cannonball travels. The optimal angle for maximum distance is 45 degrees, as this allows for the most equal distribution of horizontal and vertical components of velocity. Any angle above or below 45 degrees will result in a shorter distance traveled.

3. How does air resistance affect the motion of a cannonball?

Air resistance can significantly affect the motion of a cannonball. As the cannonball travels through the air, it will experience a force in the opposite direction of its motion due to air resistance. This force will slow down the cannonball and decrease its horizontal and vertical velocity, resulting in a shorter distance traveled.

4. What is the difference between the time of flight and the range of a cannonball?

The time of flight refers to the amount of time it takes for a cannonball to land after being fired. The range refers to the horizontal distance traveled by the cannonball before it hits the ground. These two values are related, as a longer time of flight will result in a longer range, but they are not equivalent.

5. How does the mass of the cannonball affect its trajectory?

The mass of the cannonball does not significantly affect its trajectory. As long as the other factors remain constant (such as angle of the cannon, height of the cliff, etc.), the mass of the cannonball will only affect the time of flight, not the trajectory itself. A heavier cannonball may take slightly longer to fall due to its greater inertia, but this will not change the overall trajectory of the cannonball.

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