Solve for Velocity Using Momentum Conservation

What You Will Learn

• How to apply conservation of momentum to systems with no external forces
• Understanding the difference between relative and absolute reference frames
• Converting between relative velocities using algebraic relationships
• Recognizing when the center of mass remains fixed in physics problems
• Setting up and solving momentum equations for multi-object systems

Visualizing the Problem

The girl walks right relative to the plank, but the plank recoils left to conserve momentum.

Solution: Method 1 — Momentum Conservation Approach

Step 1 — Define the system and coordinate system

Let's define rightward as positive and consider all velocities relative to the ice surface.

Step 2 — Apply conservation of momentum

Since there are no external horizontal forces on the system, momentum is conserved. The system starts at rest, so the initial momentum is zero.

Step 3 — Use the relative velocity relationship

The girl walks at 1.5 m/s to the right relative to the plank. This gives us the relationship between the two velocities.

Step 4 — Substitute and solve

Substitute the relative velocity relationship into the momentum equation.

Step 5 — Find the plank's velocity

Using our relationship from Step 3:

The negative sign indicates the plank moves leftward, which makes physical sense.

Solution: Method 2 — Center of Mass Analysis

Step 1 — Understand the center of mass principle

With no external horizontal forces, the center of mass of the system remains stationary. Since both objects start at rest, the center of mass stays fixed in space.

Step 2 — Set up the center of mass equation

If the center of mass doesn't move, then the weighted average velocity of the system is zero.

Step 3 — Express one velocity in terms of the other

From the momentum equation:

Step 4 — Apply the relative velocity constraint

The girl moves 1.5 m/s faster than the plank (rightward):

Verification

Let's check our answer by verifying momentum conservation:

We can also verify the relative velocity:

Does This Seem Reasonable?

The answer passes several sanity checks:

The girl moves slower than her walking speed: She walks at 1.5 m/s relative to the plank but only moves at 1.15 m/s relative to the ice. This makes sense because the plank recoils backward.

The plank moves backward: The plank's velocity is negative (leftward), which is exactly what we expect from Newton's third law—as the girl pushes backward on the plank to move forward, the plank pushes forward on the girl.

Mass ratio check: The plank is about 3.3 times heavier than the girl, so it should move about 3.3 times slower. Indeed, |v_p|/|v_g| = 0.35/1.15 ≈ 0.30, which is approximately 1/3.3.

Common Pitfalls

The Underlying Pattern

This problem follows a general pattern for momentum conservation with relative motion. For two objects with masses m₁ and m₂, initially at rest, where object 1 moves with velocity v_rel relative to object 2:

In our case, with m₁ = 45 kg (girl), m₂ = 150 kg (plank), and v_rel = 1.5 m/s:

This formula works whenever two objects push off each other on a frictionless surface, whether it's a person jumping off a boat, a cannon firing a cannonball, or astronauts pushing apart in space.

Where This Shows Up in Real Life

Ice skating and hockey: When two skaters push off each other, they move in opposite directions with velocities inversely related to their masses, just like this problem.

Rocket propulsion: Rockets work by ejecting mass (exhaust) in one direction, causing the rocket to accelerate in the opposite direction according to conservation of momentum.

Firearms recoil: When a bullet is fired from a gun, the bullet goes forward and the gun recoils backward. The gun moves much slower than the bullet because it's much more massive.

What If?