Preface: Some thought experiments. 
[Motion based model] Think of driving a car along a road. Your odometer is broken but your speedometer still works. You record only where the car is when you begin your trip and you make a table of the car's speed (velocity) at regular time intervals during the trip. [Or perhaps you can record the velocity at every time from the beginning to the end of the trip]. Based on this information do you think you could give your location approximately (or exactly) at any time during your travels? As a practical matter, if you have the time to repeat the trip this shouldn't be too difficult. With the information you recorded you can reproduce the trip with some accuracy. This time use a car with an odometer and let someone else do the driving according to the recorded velocities while you record your position. 

For an hour trip this might be more time than you want to spend on this. With a little thought and few arithmetic calculations, you should become up with a reasonable estimate of where you were at the same points in time where you recorded the velocities. 
 
 
Table of Velocities Read from Speedometer during Trip 
t minutes
v(t) miles/hour
0
30
15
60
30
60
45
50
60
40
  
Graph of Velocities 
Read from Speedometer during Trip 
 
A position function that matches the given instananeous velocities.
t minutes
v(t) miles/hour
Position Function s(t)
0  - 10
30
t/2
10 - 40
60
5 + (t-10)
40 - 52
50
35+5/6 (t-40)
52 - 60
40
45+2/3 (t-52)
 

Graph of Position Function based on  
given velocities from Speedometer.