Problem 30 - Calculus: Early Transcendental Functions

Question

Suppose that a runner has velocity 12 mph for 20 minutes, velocity 14 mph for 30 minutes, velocity 18 mph for 10 minutes and velocity 15 mph for 40 minutes. Find the distance run.

Step-by-Step Solution

Answer

The total distance run by the runner is the sum of distances calculated for all intervals.

Step 1: Convert Minutes to Hours

Firstly, convert the time from minutes to hours. This is due to the speed being in miles per hour. This means 1 hour is equivalent to 60 minutes. So, the time in hours is calculated by dividing each time interval by 60.

Step 2: Calculate Distance for Each Interval

Next, calculate the distance for each interval using the formula: distance = speed * time. For example, for the first interval where the runner's speed is 12 mph for 20 minutes (or 20/60 = 0.333 hours), the distance will be 12 * 0.333 = 4 miles. Repeat this calculation for all intervals.

Step 3: Sum All Distances

Finally, sum up all the distances calculated in the previous step. This will give the total distance run by the runner.

Key Concepts

Velocity and Distance

Understanding the relationship between velocity and distance is key to solving problems involving motion. Velocity, simply put, is the speed of an object in a particular direction. When you multiply the velocity by the time an object is in motion, you get the distance covered during that time period. This is based on the straightforward principle that the faster you go (higher velocity) and the longer you go for (time), the farther you will travel (distance).

Imagine running on a track; if you maintain a steady velocity, it’s easy to predict how far you will go in a certain amount of time. In our exercise, the runner changes velocity at different times, which means we need to calculate the distance covered in each segment separately and then combine them to get the total distance run.

Unit Conversion

Unit conversion is often an essential step in solving physics and math problems, and getting it wrong can lead to incorrect results. Since velocity is given in miles per hour (mph) and time in minutes, we need to convert time into hours to match the units. This ensures that when we multiply velocity by time, the resulting distance is in the appropriate unit of miles.

Unit conversion between minutes to hours is done by dividing by 60, as there are 60 minutes in an hour. It’s like converting pennies to dollars or centimeters to meters; it’s all about using conversion factors to get the right answer in terms you can use.

Distance Formula

The distance formula in the context of motion is straightforward: distance equals velocity times time, or mathematically, \( d = v \cdot t \). This formula allows you to compute the distance an object travels if you know its velocity and the amount of time it travels at that velocity.

In our example, we use this formula repeatedly to calculate the distance for each time interval given different velocities. This component-wise approach helps us manage the different velocities the runner has during the run. By applying the formula to each segment individually, we accurately account for the variations in speed and subsequently sum these distances to find the total distance covered.

Time Intervals in Motion

Time intervals play a crucial role in calculations involving motion. They represent the duration for which a particular velocity is maintained. It's important to note that when velocity changes, as it does for our runner, each segment of the run must be considered separately.

In our case, the runner's velocity changes four times, resulting in four time intervals that we must analyze individually. By being meticulous about the time spent at each velocity, we can ensure that our final distance is an accurate representation of the runner's effort. Dealing with time intervals effectively allows us to piece together the full picture of the runner's performance.