Center of Mass: Person on a Floating Raft

Center of Mass: Person on a Floating Raft

A person walks on a raft floating on frictionless water. With no external horizontal forces, the center of mass of the system stays fixed. When the person walks right, the raft drifts left — and vice versa.

Key Equations

If the person walks a distance d on the raft (relative to the raft):

Δx_raft = −m_person × d / (m_person + m_raft)

Δx_person_world = m_raft × d / (m_person + m_raft)

The raft moves opposite to the person. The total displacement of each is scaled by the other’s mass fraction. With equal masses, each moves half the distance walked.

Why Does This Happen?

• Newton’s 3rd Law: The person pushes the raft backward; the raft pushes the person forward.
• No external forces: The water is frictionless, so no horizontal force acts on the system.
• CM conservation: m_p·x_p + m_r·x_r = constant.

Try These Experiments

1. Equal Masses: Person and raft move the same distance in opposite directions. The CM is exactly halfway.
2. Light Raft: The raft moves MORE than the person walks! The floor moves out from under you.
3. Heavy Raft: Raft barely moves — like walking on solid ground.
4. Watch the graph: Switch to Time tab. The red CM line is perfectly flat — that’s conservation in action.
5. Person walks back: When the person returns to center, the raft returns to its starting position. Net displacement = 0.

Real-World Applications

• Astronaut in space station: Pushing off a wall makes both astronaut and station move.
• Gun recoil: Bullet goes forward, gun kicks backward. CM of bullet+gun stays at launch point.
• Ice skating pairs: When one skater pushes the other, both move — the lighter one faster.
• Rowing a boat: You push water backward, boat goes forward.