Unit 4 · Topic 4.1 Beta

Linear Momentum

Momentum p = mv is a vector along the velocity.

Practice 1: Creating RepresentationsPractice 2: Mathematical Routines

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Question set for this topic

Part 1 · Hook

Why this matters

A slow loaded freight train and a fast baseball are both hard to stop, for different reasons: one has enormous mass, the other a large speed. Momentum combines the two into one vector quantity, and it turns out to be the quantity collisions and explosions cannot change.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. Velocity is

  1. a vector with size and direction
  2. a scalar
  3. always positive
  4. the same as speed
Show the answer

Velocity has a direction, carried by its sign in one dimension.

  • Correct: a vector with size and direction:
  • a scalar:
  • always positive:
  • the same as speed:

2. The center of mass of a system is

  1. the mass-weighted average position
  2. the midpoint between objects
  3. the heaviest object
  4. where gravity is zero
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x_cm = Σmᵢxᵢ/Σmᵢ.

  • Correct: the mass-weighted average position:
  • the midpoint between objects:
  • the heaviest object:
  • where gravity is zero:

3. Forces between objects inside a system are

  1. internal forces
  2. external forces
  3. net forces
  4. contact forces only
Show the answer

Forces between parts of the system are internal; Newton's third law makes them cancel in pairs.

  • Correct: internal forces:
  • external forces:
  • net forces:
  • contact forces only:

Part 4 · See it

See it first

Two carts move right. Cart 1, 2.0 kg at 1.0 m/s, and cart 2, 0.50 kg at 4.0 m/s, have velocity arrows differing by four times but equal momentum arrows of 2.0 kg·m/s. Their kinetic energies are 1.0 J and 4.0 J.
Two carts with equal momentum: one heavy and slow, one light and fast, with different kinetic energies. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. An object of mass m moves with velocity vit has momentum p = mv, a vector along v, in kg·m/s
  2. Several objects form a systemthe system's momentum is the vector sum, and momenta in opposite directions cancel
  3. The system's momentum is divided by its total massthe result is the velocity of the center of mass, v_cm = Σp/Σm
  4. Objects push on each other briefly and hard (a collision or explosion)the internal forces dominate, so the system's momentum is the same just before and just after

Part 6 · Key ideas

Key ideas

  • Momentum p = mv is a vector in the direction of the velocity, in kg·m/s.
  • A system's momentum is the vector sum of its parts. Opposite momenta cancel; kinetic energies never do.
  • Same momentum does not mean same kinetic energy: K = p²/(2m), so the lighter object has more K.
  • The velocity of the center of mass is v_cm = Σp/Σm.
  • The collision model: forces between colliding objects are internal and much larger than outside forces, so the system's momentum is the same just before and just after. An explosion is the same model in reverse.

Part 7 · Misconception

A common mistake

The wrong idea: A heavy truck always has more momentum than a light ball.

What actually happens: Momentum is mass times velocity. A fast, light object can have as much momentum as a slow, heavy one, and two momenta in opposite directions can cancel.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Diagram

Two carts on level tracks

Two carts on level tracksCart 12.0 kgv = 1.0 m/sCart 20.50 kgv = 4.0 m/s

Positive is to the right.

1. What is cart 1's momentum? Give the value and its unit.

Type a number and its unit.

Show the answer

p = mv = (2.0 kg)(+1.0 m/s) = +2.0 kg·m/s, to the right. Momentum's unit is kg·m/s (the same as N·s).

  • Answer: 2.0 kg·m/s

2. How do the momenta and kinetic energies of the two carts compare?

  1. Equal momenta; cart 2 has 4 times the kinetic energy
  2. Equal momenta and equal kinetic energies
  3. Cart 2 has 4 times the momentum and 4 times the kinetic energy
  4. Cart 1 has more momentum, because it is heavier
Show the answer

p₁ = (2.0)(1.0) = 2.0 kg·m/s; p₂ = (0.50)(4.0) = 2.0 kg·m/s. K₁ = ½(2.0)(1.0)² = 1.0 J; K₂ = ½(0.50)(4.0)² = 4.0 J.

  • Correct: Equal momenta; cart 2 has 4 times the kinetic energy: Right: both p = 2.0 kg·m/s; K is 1.0 J and 4.0 J.
  • Equal momenta and equal kinetic energies: Equal momenta do not mean equal K: K = p²/(2m) is larger for the smaller mass.
  • Cart 2 has 4 times the momentum and 4 times the kinetic energy: Cart 2 is 4 times as fast but has one quarter the mass, so its momentum is the same.
  • Cart 1 has more momentum, because it is heavier: Momentum depends on mass and velocity together: 2.0 × 1.0 = 0.50 × 4.0.

3. Cart 2 reverses and moves left at 4.0 m/s. What is the momentum of the two-cart system now?

  1. 4.0 kg·m/s to the right
  2. Zero
  3. 2.0 kg·m/s to the right
  4. 5.0 J
Show the answer

Total p = (2.0 kg)(+1.0 m/s) + (0.50 kg)(−4.0 m/s) = 0. Momenta are vectors and can cancel; kinetic energies cannot.

  • 4.0 kg·m/s to the right: That adds the sizes. Cart 2 now moves left, so its momentum is negative.
  • Correct: Zero: Right: (2.0)(+1.0) + (0.50)(−4.0) = 0. Opposite momenta cancel.
  • 2.0 kg·m/s to the right: That is cart 1 alone. Add cart 2's −2.0 kg·m/s.
  • 5.0 J: That is the kinetic energy, which does not cancel. Momentum is a vector, in kg·m/s.

4. What is the velocity of the center of mass of the two carts, both moving right as shown?

  1. +2.5 m/s
  2. +5.0 m/s
  3. +1.6 m/s
  4. +4.0 m/s
Show the answer

v_cm = Σp/Σm = (2.0 + 2.0) kg·m/s / (2.0 + 0.50) kg = +1.6 m/s.

  • +2.5 m/s: This averages the two speeds. The center of mass weights each velocity by its mass.
  • +5.0 m/s: This adds the velocities. Divide the sum of the momenta by the total mass.
  • Correct: +1.6 m/s: Right: v_cm = (sum of momenta)/(total mass) = 4.0/2.5.
  • +4.0 m/s: That is the sum of the momenta; it still needs dividing by the total mass, 2.5 kg.

5. If an object's speed doubles, by what factor does its momentum change, and by what factor does its kinetic energy change?

  1. Both ×2
  2. Both ×4
  3. Momentum ×4, kinetic energy ×2
  4. Momentum ×2, kinetic energy ×4
Show the answer

p = mv doubles with v; K = ½mv² quadruples. This difference is why momentum and energy behave differently in collisions.

  • Both ×2: K depends on v², so it grows by 4.
  • Both ×4: p = mv is proportional to v, so it only doubles.
  • Momentum ×4, kinetic energy ×2: That swaps the two: p is linear in v, K quadratic.
  • Correct: Momentum ×2, kinetic energy ×4: Right: p ∝ v, K ∝ v².

6. A two-object system has a center of mass that moves at constant velocity. What can you conclude?

  1. Both objects move at constant velocity
  2. The net external force on the system is zero
  3. The two objects exert no forces on each other
  4. The system has no kinetic energy
Show the answer

The system's momentum, and so v_cm, changes only when there is a net external force. Internal forces move the parts, not the center of mass.

  • Both objects move at constant velocity: The objects can speed up and slow down as they push each other; only their combined momentum is fixed.
  • Correct: The net external force on the system is zero: Right: v_cm = Σp/Σm is constant only when the system's momentum is constant.
  • The two objects exert no forces on each other: They may interact strongly; internal forces cancel in the total.
  • The system has no kinetic energy: A moving center of mass means the system does have kinetic energy.

Part 9 · Summary

Summary

Momentum p = mv is a vector along the velocity. A system's momentum is the vector sum of its parts, and its center of mass moves at v_cm = Σp/Σm. In collisions and explosions the internal forces dominate, so the momentum just before equals the momentum just after.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections