To ward off oceanic invaders, a cannon is placed 10 meters from the water's edge, pointing out at sea. The cannon is capable of firing a projectile at a maximum velocity of 100m/s.
a) what is the maximum firing range of the cannon?
b)If the cannon was placed on a cliff that was 250m high, how much farther would the cannonball be able to travel?
To calculate the maximum firing range of the cannon, we need to consider the projectile's motion.
a) The motion of the projectile can be divided into horizontal and vertical components. In this case, we are interested in the horizontal component because we want to determine the maximum distance the cannonball can travel.
The horizontal motion of a projectile is independent of its vertical motion. This means that the time it takes for the projectile to reach the ground is the same as if it were dropped straight down from the same height.
We can use the following equation to calculate the time it takes for the projectile to hit the ground:
time = distance / velocity
In this case, since the cannon is placed 10 meters from the water's edge, the horizontal distance is our maximum range. Therefore, we have:
distance = 10 meters
velocity = 100 m/s
Plugging these values into the equation, we get:
time = 10 meters / 100 m/s
= 0.1 seconds
Now, to determine the maximum range, we multiply the time by the horizontal velocity:
maximum range = time * velocity
= 0.1 seconds * 100 m/s
= 10 meters
Therefore, the maximum firing range of the cannon is 10 meters.
b) If the cannon was placed on a cliff that was 250 meters high, it would have an additional height to provide a longer trajectory for the projectile. The vertical component of the cannonball's motion becomes significant, and we need to consider it to determine the extra distance it can travel.
To calculate this extra distance, we need to determine the time it takes for the cannonball to reach the ground when fired from the top of the cliff.
Using the vertical motion equation:
distance = (1/2) * gravitational acceleration * time^2
In this case, the distance is the height of the cliff:
distance = 250 meters
gravitational acceleration (g) = 9.8 m/s^2 (approximately)
Rearranging the equation to solve for time:
time^2 = (2 * distance) / g
time = sqrt((2 * distance) / g)
Plugging in the values, we get:
time = sqrt((2 * 250 meters) / 9.8 m/s^2)
= sqrt(500 / 9.8) s
≈ 7.10 s
Now, to determine the extended horizontal distance traveled by the cannonball, we multiply the time by the horizontal velocity:
extra distance = time * velocity
= 7.10 s * 100 m/s
= 710 meters
Therefore, if the cannon was placed on a cliff that was 250 meters high, the cannonball would be able to travel an additional 710 meters, making the total firing range 720 meters.