AP Physics C: Mechanics · Unit 4 · Topic 4 of 4 · 5 min lesson

Center of Mass Motion

A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.

Quiz this topic · 5 questions

The big idea

A system’s center of mass moves as if all its mass were there and all the external forces acted on it: F_ext = Ma_cm. Its velocity is v_cm = P_total/M. Internal forces (collisions, explosions, pushes) can’t change it.

1. The center of mass ignores internal forces

After the explosion, the pieces scatter, but their center of mass keeps following the original parabola.

v_cm = Σm_iv_i/M = P/Mvelocity of the center of mass

F_net,ext = Ma_cmNewton’s second law for systems

No external forcev_cm constant

Starts at restcenter of mass stays put

2. Using it

Collisions: v_cm is the same before, during and after, because momentum is conserved.

Explosions: track the center of mass with the original trajectory, then place the pieces around it.

Walking on a boat: with no horizontal external force, the center of mass of person + boat stays fixed. If the person moves forward relative to the water, the boat moves back: m_pΔx_p = −m_bΔx_b.

3. Example, then try it

Example

A 4.0 kg cart moves right at 2.0 m/s and a 1.0 kg cart moves left at 3.0 m/s. Find the velocity of their center of mass.

v_cm = [(4.0)(2.0) + (1.0)(−3.0)] ÷ 5.0 = 1.0 m/s right, before, during and after any collision between them.

Try it · work it on paper first

An 80 kg person walks 5.0 m toward the front of a 120 kg boat (5.0 m measured along the boat). The boat starts at rest on still water. How far does the boat move, and how far does the person move relative to the water?

Center of mass stays put: 80(person’s shift) = 120(boat’s shift), opposite directions.

Relative motion: if the boat moves back d, the person moves forward 5.0 − d relative to the water.

Solve: 80(5.0 − d) = 120d → 400 = 200d → d = 2.0 m backward.

Person: 5.0 − 2.0 = 3.0 m forward relative to the water. Check: 80(3.0) = 120(2.0) ✓.

Answer: the boat moves 2.0 m back; the person moves 3.0 m forward.

4. Common mistakes

Thinking an explosion moves the center of mass.Only external forces can.

Using the 5.0 m as the person’s displacement over the water.It’s measured relative to the boat, which also moves.

Averaging velocities without weighting by mass.v_cm = Σmv/M.

Check yourself

A rocket drifts through deep space and fires its engine. What happens to the center of mass of the rocket plus all its exhaust?

The engine force is internal to the rocket + exhaust system. With no external force, v_cm stays constant even as the rocket speeds up.

Ready to practice?

5 questions on Center of Mass Motion, each with a hint and an explanation for every wrong answer.

Quiz this topic All of Unit 4