You are a secret agent. You have ten seconds to use your secret cell phone to tell everyone where you are located.
Assume you know the vehicle has been traveling for two hours at 60 miles per hour eastward on a straight road that goes in the east direction. With what you know right now you can only say you’ve gone 120 miles.
One set of initial conditions can help you. If by some way you find out that you were at highway mile marker 80 at the start of the trip (when time=0) then you can add that 80 to the 120 and in your secret phone call you tell them that you are near mile marker 200.
Let’s examine the math:
Our formula will be d = mt + b
m = 60 miles per hour
t=2 hrs (the trip started at t=0)
We started with only having ‘m’ and ‘t’.
When we are told the initial conditions that the odometer was at 80 at the start of the trip that gave us:
Now we have:
It is typical for us to choose initial conditions that have zeroes because it was convenient for ‘mt’ to experience “death by zero”.
Bonus Feature:
There are two things to consider. Take a moment to look at how the units for every term in the equation reduced to “miles”. $$\frac {miles}{hour} miles = miles$$
This should always happen. They are tedious but units are your friends.
When we just had the part about a change of 120 miles, we had a vector and there was no coordinate system.
When we added the initial conditions we laid down a coordinate system and it was one where the origin is 80 miles behind where we were at time zero.
Vectoreverything 