Inverse Problem: Finding Velocities from Traveltimes

Travel time equation is given by 


     T  =   Dcar /Vcar    +  Dair / Vair



where

     T  <-> Travel Time  (Hours)

     Dcar  <-> Distance Traveled  by car (miles)

     Vcar  <-> Car Velocity or Speed (miles/hr)


     Dair  <-> Distance Traveled  by airplane (miles)

     Vair  <-> Velocity of airplane (miles/hr)

Last week we showed how to determine traveltimes from distance and velocity values. Now we are going to do the opposite, find velocities from traveltimes and distances.

  1. Step 1. We find it takes 24 hours to go from SLC-NY-London 

    2. 
   T  =   Dcar /Vcar    +  Dair / Vair

24 h  =  2000 m/Vcar +  4000 m/Vair 

where Vcar  = unknown car velocity from SLC-NY

      Vair = unknown airplane velocity from NY-Lond

  2. Step 2: Now find different velocities that satisfy above equation. 

    3. 
           1/Vcar      |  1/Vair
        --------------------------
             1/100     |     ?
        --------------------------
             1/200     |     ?
        --------------------------     
             1/300     |     ?
        --------------------------
             1/400     |     ?

  3. Plot up 1/Vcar versus 1/Vair and draw line. These are possible solutions. 

    4.

  4. We also know that a trip from Boston-NY-London takes 9 hours, that is 

    5. T  =   Dcar /Vcar    +  Dair / Vair

9 h  =  500 m/Vcar +  4000 m/Vair

where Vcar  = unknown velocity from Boston-NY

      Vair  = unknown velocity from NY-Lond


           1/Vcar      |  1/Vair
        --------------------------
             1/100     |     ?
        --------------------------
             1/200     |     ?
        --------------------------     
             1/300     |     ?
        --------------------------
             1/400     |     ?

    Plot up on same graph the possible values of Vcar vs Vair.

     

  5. What velocities satisfy both equations.