Momentum and Relative Velocity

[ndoc]

Homework Statement

A 56-kg woman contestant on a reality television show is at rest at the south end of a horizontal 141-kg raft that is floating in crocodile-infested waters. She and the raft are initially at rest. She needs to jump from the raft to a platform that is several meters off the north end of the raft. She takes a running start. When she reaches the north end of the raft she is running at 4.0 m/s relative to the raft. At that instant, what is her velocity relative to the water?

Homework Equations

p=person w=water r=raft
Vp = Vpr + Vrw

m*v(i) = m*v(f)

The Attempt at a Solution

Using conservation of momentum
0 = 56 * 4 + Vr*141
Solving for Vr = -1.59 m/s

Vp = 4-1.59 = 2.41 m/s which is apparently wrong and I cannot see why it would be, any help would be appreciated

 

[Andrew Mason]

The momentum of the woman + raft does not change. Therefore:

$$m_p\vec{v_{p:w}} + m_r\vec{v_{r:w}} = 0$$

where the :w means the velocity relative to the water.

To find the velocity of the woman relative to the water you have to use the relation:

$$\vec{v_{p:r}} + \vec{v_{r:w}} = \vec{v_{p:w}}$$

AM

 

[kerimek]

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I would like to clarify that the concept of relative velocity is crucial in understanding the motion of objects in relation to each other. In this scenario, the woman's velocity relative to the water is not simply the sum of her velocity relative to the raft and the raft's velocity relative to the water.

To solve this problem, we can use the formula Vp = Vpr + Vrw, where Vp is the woman's velocity relative to the water, Vpr is her velocity relative to the raft, and Vrw is the raft's velocity relative to the water.

Since the woman is initially at rest on the raft, her velocity relative to the raft is 0 m/s. Therefore, we can rewrite the formula as Vp = 0 + Vrw.

To find Vrw, we can use the conservation of momentum equation m*v(i) = m*v(f), where m is the mass of the object and v(i) and v(f) are the initial and final velocities, respectively.

In this case, the initial momentum of the system (woman and raft) is 0, since they are both initially at rest. The final momentum can be calculated by taking into account the woman's mass and velocity, as well as the raft's mass and velocity.

Therefore, we have 0 = 56*4 + 141*Vrw, which gives us Vrw = -1.59 m/s.

Substituting this value into the formula for Vp, we get Vp = 0 + (-1.59) = -1.59 m/s.

This means that at the instant when the woman reaches the north end of the raft, her velocity relative to the water is -1.59 m/s. This negative sign indicates that her velocity is in the opposite direction of the water's velocity, which is expected since she is jumping towards the north end of the raft.

In conclusion, the woman's velocity relative to the water is -1.59 m/s at the instant when she reaches the north end of the raft. It is important to remember that relative velocity takes into account the velocities of both objects in relation to a third reference point, and cannot simply be calculated by adding the individual velocities of the objects.