How Far Does the Boat Move When Juliet Joins Romeo at the Rear?

[Chuck 86]

Homework Statement

Romeo (80.3 kg) entertains Juliet (58.0 kg) by playing his guitar from the rear of their boat at rest in still water, 2.73 m away from Juliet who is in the front of the boat. After the serenade, Juliet carefully moves to the rear of the boat (away from shore) to plant a kiss on Romeo's cheek. How far does the 80.5 kg boat move toward the shore it is facing?

Homework Equations

Xcm=m1x1+m2x2+m3x3
m1+m2+m3

The Attempt at a Solution

i don't really know what I am doing on this problem because the professor didnt explain very well...at all

 

[tiny-tim]

Hi Chuck 86!

There are no (horizontal) external forces on the boat-and-Romeo-and-Juliet, so its centre of mass will stay in the same place!

 

[kerimek]

.

As a scientist, it is important to have a clear understanding of the concept of center of mass and its applications. The center of mass of a boat is the point at which the entire mass of the boat can be considered to be concentrated. In this problem, we are given the masses of Romeo and Juliet, and the distance between them on the boat. We are also asked to find how far the boat moves when Juliet moves to the rear of the boat.

To solve this problem, we can use the formula for the center of mass, which is:

Xcm = (m1x1 + m2x2 + m3x3) / (m1 + m2 + m3)

In this case, m1 represents the mass of Romeo, m2 represents the mass of Juliet, and m3 represents the mass of the boat. x1 and x2 represent the initial positions of Romeo and Juliet, respectively, and x3 represents the final position of the boat.

We can plug in the given values into the formula:

Xcm = [(80.3 kg)(2.73 m) + (58.0 kg)(0 m) + (80.5 kg)(x3)] / (80.3 kg + 58.0 kg + 80.5 kg)

Simplifying this, we get:

Xcm = (218.919 kg*m + 80.5 kg*x3) / 218.8 kg

To find the final position of the boat (x3), we can rearrange the equation to solve for x3:

x3 = (Xcm * (80.3 kg + 58.0 kg + 80.5 kg) - 218.919 kg*m) / 80.5 kg

Plugging in the value of Xcm that we calculated earlier, we get:

x3 = [(218.919 kg*m) * (218.8 kg) - 218.919 kg*m] / (80.5 kg)

Simplifying this, we get:

x3 = 218.919 m

Therefore, the boat moves 218.919 m towards the shore it is facing when Juliet moves to the rear of the boat to kiss Romeo. This is because the center of mass of the boat and its occupants shifts towards the rear of the boat, causing the boat to move in the opposite direction.