What is the new distance between the man and the shore after he switched seats?

[mrnastytime]

Homework Statement

A man decides to boat out on the lake for a relaxing day of fishing. At one point during the day, he finds himself facing the shore. He judges that he is a distance Dinit = 35 ft from the shore. The man has a mass of Mman = 190 lbs, and the boat a mass of Mboat = 270 lbs. The boat has two seats each located a distance x = 5 ft from the center of the boat.

Homework Equations

Where is the center-of-mass of the man-boat system relative to the shore?
Deciding that more fish are to be found closer to shore, the man switches to the other seat. Assume there is no friction between the boat and the water. After switching seats, the man realizes that the boat has moved farther from the shore.

What is the new distance between the man and the shore after he switched seats?

The Attempt at a Solution

Xcm=X1*M1+X2*M2/(M1+M2)
Xcm=35*190+(35-5)*270/(190+270)
This is not the correct answer. What equation am i suppose to use?

 

[mrnastytime]

nevermind i figured it out

 

[nlightner]

To find the center of mass of the man-boat system, you can use the equation:
Xcm = (Mman*Xman + Mboat*Xboat) / (Mman + Mboat)
Where Xman is the distance of the man from the center of the boat (5 ft in this case) and Xboat is the distance of the boat from the center of the boat (which is half the distance between the two seats, or 2.5 ft). Plugging in the values, we get:
Xcm = (190*5 + 270*2.5) / (190+270) = 3.75 ft
This means that the center of mass of the man-boat system is 3.75 ft from the center of the boat towards the man's original seat.
To find the new distance between the man and the shore after switching seats, we can use the equation:
Dnew = Dinit + (Xcm - Xman)
Where Dinit is the initial distance between the man and the shore (35 ft in this case), Xcm is the center of mass of the man-boat system (3.75 ft) and Xman is the distance of the man from the center of the boat (5 ft). Plugging in the values, we get:
Dnew = 35 + (3.75 - 5) = 33.75 ft
Therefore, the new distance between the man and the shore is 33.75 ft. This makes sense because when the man switches to the other seat, the center of mass of the man-boat system shifts towards his original seat, causing the boat to move further away from the shore.