STUDY TOOL FOR THE STUDENTS WHO SEARCH NOTES BY ONLINE, Lecture notes of Earth science

Download STUDY TOOL FOR THE STUDENTS WHO SEARCH NOTES BY ONLINE and more Lecture notes Earth science in PDF only on Docsity! The Velocity Problem Investigating Calculus: The Velocity Problem Let's start our exploration of calculus by examining what I like to call the velocity problem. Imagine I'm driving on the highway and I state that at 2:00 o'clock I'm at the hundred mile marker, but at 2:15 I will be at the hundred and ten mile marker. The question is, how fast am I going? However, to answer this question, we need to be more precise about what we're asking. Average Velocity One way to answer this question is by determining the average velocity. The average velocity is calculated by dividing the change in distance by the change in time. Mathematically, this can be represented as delta v (change in distance) divided by delta t (change in time). Let's compute the average velocity over the 15-minute time interval we have. The change in distance (delta d) is 110 miles minus 100 miles, which equals 10 miles. The change in time (delta t) is 2:15 minus 2 o'clock, which equals 15 minutes. However, speeds are typically given in miles per hour, not miles per minute. To convert the units, we can multiply the minutes by 60 to get the number of minutes in an hour. This gives us 600 divided by 15, which equals 40 miles per hour. Therefore, the average velocity over this 15-minute interval is 40 miles per hour. Instantaneous Velocity Now, let's consider a slightly different question: What would a cop measure at exactly 2:15? Previously, we determined that the average velocity over the 15- minute interval was 40 miles per hour. However, this does not provide enough information to answer the question of the cop's measurement at 2:15 precisely. For example, let's say I was speeding from 2 o'clock until 2:10, but then I encountered a traffic light and slowed down. The cop might not measure my higher speed because the average velocity over the entire interval is still 40 miles per hour. However, at some points during that interval, I may have been traveling much faster or slower than 40 miles per hour. What we're really asking here is the instantaneous velocity at one exact time, which is different from the average velocity over an interval. To better understand this concept, let's consider a table of data with increasingly smaller time intervals. Chart of Average Velocities  2:00 to 2:15: 40 miles per hour  2:00 to 2:14: 45 miles per hour  2:00 to 2:13: 48 miles per hour  2:00 to 2:12: 50 miles per hour  ...  2:00 to 2:01: 57 miles per hour  2:00 to 2:00:59: 59 miles per hour  2:00 to 2:00:58: 59.5 miles per hour  2:00 to 2:00:57: 59.7 miles per hour  ...  2:00 to 2:00:01: 59.9 miles per hour  2:00 to 2:00:00: 60 miles per hour