Chukat- Two Approaches to Distance Traveled as Related to Rate of Speed

This week I would like to follow-up on my post from Parshat Chukat last year. In last year's post, I looked at the distance that the Children of Israel traveled in Bamidbar Chapters 20-21. Using an estimated route, I calculated that they traveled approximately 312.5 miles in this final section of their wandering in the desert.

"The Children of Israel, the entire assembly, arrived at the Wilderness of Zin in the first month and the people settled in Kadesh. Miriam died there and she was buried there." ~Numbers 20:1

From Midrash, we learn that these two chapters take place in the final year of the wandering. We learn in this first statement that we are beginning this section from the first month of this final year of wandering. Throughout the course of these 2 chapters, we have Miriam's death with 30 days of mourning and Aaron's death with 30 days of mourning. This means that out of the final full year of wandering, the Children of Israel had at most 10 months of travel time. While we know that there were periods of traveling and periods of encampment, we can still calculate an average rate of travel in terms of miles traveled per month. We could break it down further to calculate average miles traveled per day or even per hour. Let's remember that when you look at an average rate, it doesn't mean that you are traveling consistently over the whole time period, but if you broke up your entire travel so that you were traveling equal distances each hour, day, week, or month, this is how much you would have traveled in that space of time.

Back in Parshat Vayetzei, I first looked at Rate of Speed, where you express how fast or slow something or someones covers a distance by comparing the distance covered to the amount of time that it takes to cover the distance. This comparison is expressed as a ratio. So, if I travel 30 miles over the course of 1 hour, then my rate of speed would be 30 miles per hour (30mi/1hr or 30mph). If I travel 100 miles in 2 hours, then my rate could be expressed as 100mi/2hr or reduced as 50mph (since 100/2 reduces to 50/1).

Using this understanding of rate of speed, how could we look at the travel speed of the Children of Israel during the end of their wandering in the desert?

Rate of travel per month: 

Based on the estimates that I calculated last year, the distance traveled was 312.5 miles, and using my timeline logic above, they had 10 months of potential travel time.

Therefore, the rate per month would have been: 

312.5mi/10mo

This reduces to an average of 31.25mi/1mo or 31.25 miles per month

Follow-up:

Following this thinking-

*you can compare how many days would have been in the 10 months of travel time, and then calculate the estimated rate of miles per day

*how many hours would have been in those 10 months? what would the average rate of miles per hour have been?

*would the miles per hour calculation be a realistic estimate? Think about what this would mean for travel time and rest time during each day or week. How might you adjust this calculation to account for resting time each day (don't forget stopping for Shabbat)? How might you adjust the original calculations to allow for estimated periods of encampment based on the timeline that we're given in the parsha?

An alternate approach:
Looking at it from a different angle, if we estimate a reasonable rate of speed for their travel, how many hours/days/weeks/months would they have needed to be traveling and how much time would they have had for their periods of encampment?

*What does this mean? What would be a reasonable estimate for the amount of distance they were able to travel in an hour? a day? a week? a month? By quantifying some of these rates, how might their travel have lined up with the information that is provided in the parsha? How long would it have taken them to get between their different stops where they requested and were denied permission to cross through other lands? How might you create an estimated timeline of their travel using all of this information?

*Personal connection: Think about it similar to planning a road trip- how much mileage do you have to cover? What stops or activity locations might you be planning, and what are the distances between the stops? What is a reasonable rate of speed for your travel? How long will it take you between each stop? Can you fit your trip within a given timeframe or do you need more time to reasonably make your full trip?

Note: It is important to remember that this information is all hypothetical, since we don't have day by day or month by month itemizations of travel time or encampment periods. Additionally, my calculations from last year were merely estimates as well. That being said, thinking about the situation within this context can help students create a semblance of understanding around what the Children of Israel might have actually been going through in their travels- offering a realistic link to this "story".

Vayetzei- Rate of Speed: What's your walking speed?

"And [Lavan] put a distance of three days between himself and Jacob; and Jacob tended Lavan's flock that remained" ~Bereishit 30;36

"So [Jacob] fled- he and all he had; he arose and crossed the river and he set his direction toward Mount Gilead. It was told to Lavan on the third day that Jacob had fled. So he took his brethren with him and he chased after him a seven-days' journey, and caught up with him on Mount Gilead." ~Bereishit 31;21-23

Rashi here explains the timing and distance. To begin with, Lavan was already 3 days away from Jacob, as we are told in 30;36. Therefore, when Jacob left, it took the informer 3 days to reach Lavan with the news of Jacob's departure. Rashi then explains that "a seven-days' journey" is for us to understand that, while Jacob was 7 days ahead of Lavan (3 days to begin with, plus 3 days of travel forward while the informer was traveling in the other direction to tell Lavan, plus the day on which Lavan reached him), Lavan actually covered this distance to reach Jacob on his first day of travel- 4 days after Jacob left.

Rate: a quantity, amount, or degree of something measured per unit of something else (merriam-webster.com)

Basically, when comparing any two different measurements to each other, you have a rate. Common examples from our daily lives are miles per hour (speed), or dollars per pound (cost). When calculating with a rate, it is helpful to calculate a base rate, which tells you the amount of one measurement per one single unit of the other measurement.

If we look at how far Jacob and Lavan each traveled, and the length of time that it took each of them to travel their respective distances, then we can calculate the rate of speed that each of them was traveling. 

From the text and Rashi's commentary, we know that Jacob traveled for a total of 4 days, and Lavan traveled for a total of 1 days. Now for the distance that each man traveled:

Based on the information that we have from the text and from Rashi, let's assume that the distance that they were each traveling was from the general Padan-Aram/Haran area to Mount Gilead. On a current map, we can use the map coordinates for Harran and for Ramoth-Gilead to approximate the distance that they traveled. Although it won't calculate travel distance, if we plot the coordinates onto a map and estimate the distance between the two locations, it's approximately 300 mi between the two locations. 

Now, let's consider that Jacob (and his wives and children and sheep) traveled those 300 miles in 4 days, and Lavan chased after him and traveled the 300 miles in 1 day. 
7-days journey- If the Torah assumes that this journey should have taken 7 days, then 300 miles in 7 days is roughly 43 miles each day. Assuming 12 hours of walking time each day, the standard travel rate would be about 3.5 m/h (miles per hour).
Jacob- 300 miles was the 7 day journey. If Jacob was only traveling for 4 days (note that he was 3 days away from Lavan's flocks), we can subtract about 120 miles from his trip (3 days x 40 miles each day). This means that he traveled 300-120= 180 miles. These 180 miles divided into 4 days means that he traveled 45 miles per day. This is equal to just under 2 m/h if he didn't stop to rest or sleep, but just walked 24 hours a day straight for the 4 days. To be more realistic, let's assume that he walked a total of 12 hours each day- to allow for sleeping at night and resting with children. 45 miles divided into 12 hours means that they were walking at a leisurely rate of 3.75 m/h.
Lavan- He traveled 300 miles in a single day. Since we know that he was chasing after Jacob to catch him right away, and he was running with just his brothers, let's assume that they ran until they caught up to Jacob. If it took him a full 24 hours, then he was running at a rate of 12.5 m/h. If we assume that he needed to stop at nightfall, and could, at most, travel 12 hours in the day, then he was traveling at a rate of 25 m/h. 

Before you scoff at these rates, we could say that they were actually traveling by camel (it does specifically say this for Jacob- Bereishit 31;17), which is able to reach speeds of 40 m/h and sustain a speed of up to 25 m/h (ref. wikipedia).