Expert: Chanda Walker - 3/30/2009

Question
An heiress is throwing a party on her yacht.  She is in her room till 12 o'clock. at twelve thirty she find that her jewels are missing. Col. Mustard is missing from 12:00-12:10. Mrs. Peacock is missing from 12:10-12:20 and the butler is missing from 12:20-12:30. at 1:30 the boat is turned around to inform authorities.  at 2:45 the jewels are found floating in a jar in the water. Without knowing the speed of the boat or the current which is going in the direction of the boat after it turns around.  who stole the jewels and when were they thrown overboard. Please help I do not understand how to do a problem like this without more numbers thanks.

Answer
All we know is "the current which is going in the direction of the boat after it turns around."

Let's name some variables:
v = velocity of the boat
c = velocity of the current
d = distance of boat to the jewels (a function of time)
t = time in min from 12:00
tt = time of theft

Basic equations needed:

v*t = d

Problem set up:
d = (v - c)*(t - tt) + c*(t - tt)  for t<90
dmax = (v - c)*(90 - tt) + c*(90 - tt)  (t = 90)

simplifying  . . .

d = v*(t - tt)
dmax = v*(90 - tt)

NOTICE!!!  Even though the current slows down the boat, it speeds up the jewels so it cancels out of this equations.  That will be helpful!



d = dmax - (v + c)*(t - tt -90) + c*(t - tt -90) for t>90
dfound = dmax - (v + c)*(90 - tt) + c*(90 - tt)   (t = 180)

simplifying . . .
d = dmax - v*(t - tt)-c*(t-tt)+c*(t-tt) + v*90 + c*90 - c*90
d = dmax - v*(t - tt) + v*90

Again, the current has no influence on the distance between the jewels and the boat.  Only the velocity matters.

So we can assume the current is zero (even if it is not.)  This will make the problem easier to wrap our brain around.

If the current is zero, the jewels are just exactly where they were dropped.  So the time to travel away is exactly equal to the time to travel back.

Let's convert time into minutes so you are sure to follow my math.

12:00 = 0 min
12:30 = 30 min
1:30 = 90 min
2:45 = 165 min

The distance from the boat at maximum distance (2:45) to the jewels at the turn around point (1:30) is:

d = (165 - 90)*v
d = 75v

Starting from the time the jewels were stolen, tstolen, we traveled for d/v minutes and at that time it was 1:30.

So
tstolen + d/v = 90
tstolen = 90  - d/v
tstolen = 90 - 75v/v
tstolen = 90 -75
tstolen = 15

That means that the jewels were stolen at 12:15 and Mrs. Peacock is under arrest.  And she looks so sweet.

BTW, when you do this problem in a physics class, you can ignore the current from the onset because you put the problem in the current's frame of reference.  I wasn't sure that you could "pull that trick" so I proved it for you.