Conservation of momentum of a boat

In summary, the man walking across the 4m long, 200kg boat will cause the center of mass to shift 1.04m from its starting position.
  • #1
Chiara
can u please tell me what i did wrong?
A 70 Kg person stands at the back of a 200 Kg boat of length 4 m that floats on stationary water. he begins to walk toward the front of the boat. When he gets to the front how far back will the boat have moved? (neglect the resistence of water)

The initial momentum is 0 both for the person and the boat since they are still. As the person begins to walk his momentum changes by an amount proportional to the force exerted on the boat to move forward:
§p=F*§t where §=change
p= momentum
t= time
§p=(70)(9.8)*§t
this §p has to be equal and opposite in sign to the §p of the boat for the law of conservation of momentum.
(70)(9.8)*§t=(200)a*§t where a is the acceleration of the boat.
a=3.43m/s2
now we can calculate the relationship between the space the person moves through and the space the boat moves
s=Vt+(1/2)at^2 that is, since the initial velocity is0 s=(1/2)at^2
s(person)=(1/2)(9.8)§t^2
s(boat)= (1/2)/(3.43)§t^2
therefore s(person)/s(boat)= 2.86
2.86=4m/(total s of the boat)
total distance the boat moves=1.4 m
the answer given in the book is 1.04 and not 1.4
Please, tell me what I did wrong!
 
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  • #2
Hi,

You have not considered the Centre of Mass of the system.
Consider the mass of the man to be m, mass of the boat to be M,
velocity of the man to be u and velocity of the centre of mass of the
man-boat system to be v.

The velocity of the boat gets adjusted to the velocity of the man walking
on the boat.

Therefore,

(M+m)*v = m*u.

Thus,

u = v*(1+(M/m)). Calculate the time taken for the man to walk the
length of the boat (4m) = t.

In this time the boat travels a distance of (v*t) m.

Note: use the ratio of u/v in order to solve the problem (since the value of the
boat velocity is not given)

Hope u got ur solution...


Sridhar
 
  • #3
One major mistake you made was in calculating the force as the force of gravity. The man is walking horizontally, not vertically!
Since the force applied would depend upon the resistance of the water, which is not given, I don't see any good way of calculating the force applied.

sridhar_n's suggestion- Calculating the center of mass both before and after the motion- is what I would recommend. Since there is no external force, the center of mass should remain fixed.
 
  • #4
I know I'm probably a few months late and I doubt you need the answer anymore, but I'll post this in case someone else comes across a similar problem.

You were going in the wrong direction with your solution. This is not a conservation of momentum problem; it is a center a mass problem. HallsOfIvy was correct to say that since there are no external forces acting on the system, the center of mass remains the same. Therefore, you need to determine how the man walking across the boat changes the center of mass, and how the boat will move to compensate.

To calculate the center of mass of a system:

(m1*x1 +...+ mn*xn) / M

where n is the number of particles within the system and M is the total mass of the system.

The center of mass in your problem is (4m*70kg) / (200kg+70kg) = 1.04m from 0, the starting position. The boat therefore moves 1.04m.
 

1. What is the conservation of momentum of a boat?

The conservation of momentum of a boat refers to the principle that the total momentum of a system of objects, in this case a boat and its surroundings, will remain constant unless acted upon by an external force.

2. How does the conservation of momentum apply to a boat?

In the context of a boat, the conservation of momentum means that the total momentum of the boat and the water it displaces will remain constant, even as the boat moves through the water.

3. Why is the conservation of momentum important in boating?

The conservation of momentum is important in boating because it helps to explain and predict how a boat will behave in the water. It also allows us to understand the forces at work when a boat is accelerating or decelerating.

4. Can conservation of momentum be violated in boating?

No, conservation of momentum is a fundamental principle of physics and cannot be violated. However, in practical situations, it may appear as though momentum is not conserved due to external forces such as wind or waves.

5. How can conservation of momentum be applied in boat design?

Boat designers can use the principle of conservation of momentum to optimize the design and performance of a boat. By understanding the relationship between a boat's momentum and the water it displaces, designers can create more efficient and stable boats.

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