Aim for FivePhysics C: Mechanics

Unit 4 · Topic 4.1

4.1 Linear Momentum

Linear momentum, p⃗=mv⃗\vec{p} = m\vec{v}, measures how much motion an object carries, combining its mass and velocity. It's a vector, so momenta in opposite directions can cancel. Momentum is the best tool for collisions and explosions, where the forces between objects are huge, brief and hard to measure.

Key terms

  • linear momentum
  • vector
  • collision
  • explosion
  • object model

What momentum is

An object's linear momentum is its mass times its velocity:

p⃗=mv⃗\vec{p} = m\vec{v}

Its unit is the kg·m/s (which equals a N·s). Because mass is positive, momentum always points the same way as the velocity. A 0.15 kg baseball at 40 m/s has a momentum of 6.0 kg·m/s; a 1500 kg car creeping at 0.004 m/s has the same.

"Linear" separates it from angular momentum, the momentum of spinning, which comes in Unit 6.

Momentum is a vector

The total momentum of a system is the vector sum of its parts' momenta: p⃗sys=∑miv⃗i\vec{p}_{sys} = \sum m_i\vec{v}_i. In one dimension, use signs for direction. Two carts with equal mass moving toward each other at equal speeds have zero total momentum, even though both are moving.

In two dimensions, add momenta by components. Momentum along x and momentum along y are tracked separately.

Compare kinetic energy, a scalar that never cancels. Those two carts with zero total momentum still have plenty of kinetic energy. That difference is why momentum and energy give separate, independent equations in collision problems.

When momentum is the right tool

During a collision (two objects hitting) or an explosion (parts of a system pushed apart by internal forces), the forces between the objects are enormous and last only a very short time. They're hard to know in detail, but they come in third-law pairs, so they can't change the system's total momentum.

External forces like gravity or friction are usually much smaller than those collision forces, and they act for only that brief moment. So their effect on momentum during the collision is negligible, and you compare the system's momentum just before and just after.

Because you only compare those two moments, each object can be modeled as a single point with its mass at its center of mass. Its shape and what happens inside it during the crash don't matter.

Momentum and kinetic energy

Momentum and kinetic energy are linked: K=p22mK = \frac{p^2}{2m}. For the same momentum, a lighter object has more kinetic energy. When a rifle fires a bullet, the bullet and rifle get equal and opposite momenta, but the light bullet carries almost all the kinetic energy, which is why the bullet is dangerous and the rifle's kick is manageable.

Worked examples

Try each one yourself first, then open the solution.

  1. Example 1

    Total momentum versus total kinetic energy

    A 2.0 kg cart moves right at 3.0 m/s and a 1.5 kg cart moves left at 4.0 m/s. Find the system's total momentum and total kinetic energy.

    Show the solution
    1. Step 1: Take right as positive. Momenta: (2.0)(3.0) = +6.0 kg·m/s and (1.5)(−4.0) = −6.0 kg·m/s.
    2. Step 2: Total momentum: 6.0 + (−6.0) = 0.
    3. Step 3: Kinetic energies: 12(2.0)(3.0)2=9.0\frac{1}{2}(2.0)(3.0)^2 = 9.0 J and 12(1.5)(4.0)2=12\frac{1}{2}(1.5)(4.0)^2 = 12 J. Total: 21 J.
    4. Step 4: The trap is thinking zero total momentum means nothing is moving or there's no energy. Momentum can cancel; kinetic energy can't.

    Answer: Total momentum 0; total kinetic energy 21 J.

  2. Example 2Calculator allowed

    Momentum in components

    A 0.50 kg puck slides at 6.0 m/s in a direction 30° north of east. Write its momentum in unit vector notation (east = +x, north = +y).

    Show the solution
    1. Step 1: Magnitude: p=(0.50)(6.0)=3.0p = (0.50)(6.0) = 3.0 kg·m/s, in the direction of the velocity.
    2. Step 2: px=3.0cos⁡30∘≈2.6p_x = 3.0\cos 30^\circ \approx 2.6 kg·m/s and py=3.0sin⁡30∘=1.5p_y = 3.0\sin 30^\circ = 1.5 kg·m/s.

    Answer: p⃗≈(2.6i^+1.5j^)\vec{p} \approx (2.6\hat{i} + 1.5\hat{j}) kg·m/s.

Common mistakes

  • Adding momentum magnitudes without signs or components. Momentum is a vector.
  • Treating momentum and kinetic energy as interchangeable. One is a vector that can cancel; the other is a scalar that can't.
  • Including the details of an object's shape in a collision. For before-and-after comparisons, each object is a point.

On the exam

  • Expect comparisons of two objects' momenta and kinetic energies, such as a heavy slow object and a light fast one. Use K=p22mK = \frac{p^2}{2m} to compare quickly.
  • When asked for a momentum, give a direction (or a sign with your stated positive direction) as well as a size.

Connected topics

Videos

  • Topic 4.1 - Linear Momentum

    Lessons With LondotWatch on YouTube (opens in a new tab)

  • Momentum | AP Physics | Khan Academy

    Khan AcademyWatch on YouTube (opens in a new tab)

  • You Can't Run From Momentum! (a momentum introduction)

    Flipping PhysicsWatch on YouTube (opens in a new tab)

  • Linear Momentum

    Bozeman ScienceWatch on YouTube (opens in a new tab)

  • 15.4 Momentum of a System of Point Particles

    MIT OpenCourseWareWatch on YouTube (opens in a new tab)

  • Introduction to Momentum, Force, Newton's Second Law, Conservation of Linear Momentum, Physics

    The Organic Chemistry TutorWatch on YouTube (opens in a new tab)

Check yourself

4 questions on 4.1 Linear Momentum. Pick an answer to see if you got it, and why.

Question 1 of 4Calculator allowed

A 0.15 kg baseball moves at 40 m/s toward home plate. What is the magnitude of its momentum?

Question 2 of 4Calculator allowed

A 2.0 kg puck slides at 5.0 m/s in a direction 37° above the +x-axis (sin 37° = 0.60, cos 37° = 0.80). What are the components of its momentum?

Question 3 of 4Calculator allowed

A bowling ball and a tennis ball have the same kinetic energy. Which has the greater momentum?

Question 4 of 4Calculator allowed

A truck and a bicycle have momenta of the same magnitude. Which has more kinetic energy?

0 of 4 answered