The standard 26 -mile, 385 -yard marathon dates to \(1908,\) when the Olympic marathon started at Windsor Castle and finished before the Royal Box at London's Olympic Stadium. Today's top marathoners achieve times around 2 hours, 4 minutes for the standard marathon. (a) What's the average speed of a marathon run in this time? (b) Marathons before 1908 were typically about 25 miles. How much longer does the race last today as a result of the extra mile and 385 yards, assuming it's run at the average speed?

Understanding the concepts of distance and displacement is crucial for solving problems in physics, particularly when calculating average speed. Distance refers to the total path length traveled by an object, regardless of its direction, and is a scalar quantity, which means it only has magnitude and no direction.

Displacement, on the other hand, is the shortest path between the initial and final points of an object's journey and is a vector quantity. This means that displacement has both magnitude and direction. While both concepts measure a certain 'length', in physics, they are used in different contexts. For instance, if a marathon runner completes a circular track of 26 miles, their distance traveled is 26 miles, but their displacement could be zero if they start and finish at the same point.

In calculating average speed for a marathon with a known length, distance is used rather than displacement, as the path taken by the runners does not return to the start point. Therefore, in a marathon, we focus on the total ground covered, which is the distance.

In physics, as well as in many other fields, unit conversion is an essential skill that ensures measurements are accurate and comparable. Different units are used for measuring distance, such as yards, miles, meters, and kilometers. Being able to convert from one unit to another can sometimes make or break a problem's solution.

For example, the additional length of a marathon (385 yards) must be converted to miles because the other part of the measurement is already in miles (26 miles). We use the conversion factor where 1 mile is equivalent to 1760 yards to maintain consistency and to be able to calculate the correct average speed.

Conversion Process

Here's how the conversion is done:

• First, understand the relationship between the units. In this case, 1 mile = 1760 yards.
• Then, divide the number of yards by the conversion factor: 385 yards / 1760 yards/mile = 0.219 miles.

Now, we can accurately add 0.219 miles to 26 miles which gives us the total marathon distance used in calculations.

Time calculation is a fundamental part of physics that involves determining the duration of events. In the context of average speed and distance, the time taken to complete an activity plays a key role.

Calculating time becomes vital, especially when comparing historical data with current records as in the marathon example. Here, we use the fact that average speed equals distance divided by time taken to solve for time. The formula is rearranged to solve for time when distance and speed are known: \(Time = \frac{Distance}{Speed}\).

When the marathon distance increased, it became necessary to calculate how much additional time a runner would need to complete the new longer distance, assuming they are running at the same average speed. By applying the time formula with the found average speed and the additional distance, we can determine the extra time required for the current marathon distance. This illustrates the real-world applicability of these physics concepts and highlights the importance of proper time calculation in understanding how changes in distance or speed can affect outcomes.