In a fit of pique, the Canadians fire a cannonball due south from vancouver (lat 49.2 degrees N) at a randomly picked city (39.4 degrees N). Assuming the cannonball takes 48 minutes to get there, by how far and in what direction will the cannonball miss?
How much they miss by depends upon whether they used the right combination of alunch angle and velocity. You have said nothing about what they were.
I suspect that they want to to compute the error due to Coriolis acceleration, assuming that they calculated the target correctly for a non-rotating Earth. Let a' be the Coriolis acceleration, which depends upon latitude. Use a midflight average value of 44.3 degrees. The target error will be
(1/2)a' t^2 , where t is the time of flight in seconds.
To determine how far and in what direction the cannonball will miss, we can use basic trigonometry and geography. Here's how you can calculate it step by step:
Step 1: Determine the latitude difference between Vancouver (starting point) and the random city (destination). In this case, the difference is 49.2 - 39.4 = 9.8 degrees.
Step 2: Convert the latitude difference from degrees to nautical miles. Since 1 degree of latitude is approximately 60 nautical miles, we multiply the latitude difference by 60. Thus, 9.8 degrees * 60 nm/degree = 588 nautical miles.
Step 3: Calculate the time difference. The cannonball takes 48 minutes to reach the destination city.
Step 4: Determine the speed of the cannonball. To calculate this, divide the distance (588 nautical miles) by the time (48 minutes) to get miles per minute. Since there are approximately 1.15078 statute miles in a nautical mile, divide the result by 1.15078 to convert it to statute miles.
Thus, the speed of the cannonball is (588 nm / 48 min) / 1.15078 = 10.05 statute miles per minute.
Step 5: Determine the distance the cannonball would have traveled if it had traveled directly south for 48 minutes at the calculated speed. Multiply the speed (10.05 miles/minute) by the time (48 minutes) to get the distance. In this case, it is 10.05 miles/minute * 48 minutes = 482.4 miles.
Step 6: Calculate the difference between the actual distance traveled (482.4 miles) and the intended distance traveled (straight south for 48 minutes). This will give us the distance by which the cannonball misses its target.
In conclusion, the cannonball will miss the target by approximately 482.4 - 588 = −105.6 miles, to the east or west depending on the chosen direction (+ for east, - for west).