if you shoot a cannonball due north from a latitude of -45o where will it fall 100km from the cannon
At a more northern latitude, but still in the southern hemisphere. Figure out how many degrees of longitude correspond to 100 km and add it to -45. It's roughly one degree.
Is this really what they are considering astronomy these days?
I should have said degrees of latitude, not longitude.
One degree of either is about 111 km.
100 km is therefore 0.901 degrees
To determine where the cannonball will fall, you need to consider the Earth's rotation and the projectile's initial velocity and angle. Here's how you can calculate it:
1. Convert the latitude to radians: -45 degrees = -45 * pi/180 radians ≈ -0.7854 radians.
2. Determine the velocity of the cannonball. Assume an initial velocity of "v" m/s.
3. Split the velocity into its northward and eastward components. Since we are shooting due north, the eastward component will be zero.
4. Calculate the northward component of the velocity using trigonometry. The northward component is given by v * cos(latitude). Substitute the latitude in radians to get v * cos(-0.7854).
5. Use the kinematic equations to calculate the time the cannonball will be in the air. The vertical motion of the cannonball can be treated independently, as there is no acceleration in the east-west direction. The equation to calculate time is t = (2 * vertical displacement) / vertical velocity, assuming the cannonball falls back to the same level.
6. Calculate the vertical displacement. Since we want the cannonball to fall 100km (100,000 meters) from the cannon, the vertical displacement will be -100,000 meters (negative because it falls).
7. Use the time calculated in step 5 and the northward component of velocity calculated in step 4 to determine how far north the cannonball will travel. Multiply the time by the northward velocity to get the distance traveled north.
8. The distance north will give you the answer to where the cannonball will fall, as it will fall directly north from the starting point.
Note: This calculation assumes ideal projectile motion without considering air resistance or the Earth's curvature. It also assumes the cannonball is not affected by any external forces during its flight.