Conservation of Linear Momentum — Question 2

Question 2

A walker shifts the boat beneath them

A $60\,\mathrm{kg}$ person walks $3.0\,\mathrm m$ to the right relative to a $120\,\mathrm{kg}$ boat. Initially both are at rest. Ignore horizontal water forces; no mass leaves the system.

1. Find boat and person displacements relative to shore.

2. If the walking speed varies, do the final displacements change?

Question 2 - Solution

A walker shifts the boat beneath them

(a) Derivation and calculation

The isolated system begins with zero momentum, so its center stays fixed. Let the boat displacement be $b$: $$\Delta x_p=b+3,\qquad120b+60(b+3)=0.$$ $$\boxed{b=-1.0\ \mathrm m},\qquad\boxed{\Delta x_p=2.0\ \mathrm m}.$$

(b) Result and interpretation

The fixed-center constraint holds at every instant: $$120v_b+60v_p=0\Rightarrow120\Delta x_b+60\Delta x_p=0.$$ Only the final relative displacement enters the integrated relation. Varying the walking speed changes the timing but not these final displacements.

Physical check

The person moves 2 m relative to shore while the boat moves 1 m oppositely, giving 3 m relative travel.