Interactive module First-year engineering dynamics

Impulse and Momentum

How hard does a bat hit a ball in a millisecond of contact? How long does a car take to stop on a grade, how fast do two railcars roll after they couple, how high does a ballistic pendulum swing? Impulse and momentum answer these by integrating \(\sum\Fvec = m\avec\) over time, once and for all. Eight short lessons, with animated figures and questions that check your work, take you from the impulse of a force to collisions at an angle.

Start Lesson 1
  • 8 lessons
  • 270 min of lessons
  • Works offline
Two carts collide on a smooth track and bounce apart. Watch the bars: each cart's momentum \(\colL{mv}\) changes in the impact, by equal and opposite amounts, and their total never moves. Lesson 6 finds the velocities afterward.

Learning outcomes

By the end of the module you can:

  • Explain how integrating \(\sum\Fvec = m\avec\) over time relates velocity to time, and choose impulse and momentum for problems that link the two. (Lesson 1)
  • Calculate linear momentum \(m\vvec\) and the impulse of a force, as \(\int\Fvec\,dt\) and as the area under its force–time graph, and find average forces. (Lesson 2)
  • Apply the principle of linear impulse and momentum, \(m\vvec_1 + \sum\int\Fvec\,dt = m\vvec_2\), in components, to find velocities, times and forces, including forces that vary with time. (Lessons 3–4)
  • Decide when the momentum of a system is conserved, using the difference between impulsive and nonimpulsive forces, and apply it to bodies that push apart or join. (Lesson 5)
  • Find the velocities after a direct central impact from conservation of momentum and the coefficient of restitution, and the kinetic energy lost. (Lesson 6)
  • Analyze oblique impact with normal and tangential axes, for a ball on a fixed surface and for two bodies. (Lesson 7)
  • Chain impact, momentum and energy in problems with several stages, such as an impact into a spring or a ballistic pendulum. (Lesson 8)

How to use this module

Before you start: you should be comfortable with free-body diagrams and \(\sum\Fvec = m\avec\), vector components, integrals of simple polynomials and sines, and the principle of work and energy (the Work and Energy module), which Lesson 8 combines with impact. The module uses the notation of Hibbeler's Engineering Mechanics: Dynamics (\(\Lvec = m\vvec\), \(\int\Fvec\,dt\), \((v_A)_1\), \((v_A)_2\), \(e\)); Beer & Johnston and Meriam & Kraige use the same ideas with small differences in notation, pointed out where they occur.

Work through the lessons in order. The times allow for working the examples on paper and doing the checks. A suggested plan in four sittings:

  1. Lessons 1–390 min

    Why impulse and momentum, linear momentum and the impulse of a force, and the principle of impulse and momentum.

  2. Lessons 4–570 min

    Forces that vary with time, then systems of particles and the conservation of momentum.

  3. Lessons 6–8110 min

    Direct central impact and the coefficient of restitution, oblique impact, and problems that combine impact, momentum and energy.

  4. Practice Lab and Self-Check Quiz65 min

    About 30 min of practice on the topics you found hardest, then the 35-minute quiz.

About 5½ h in total:270 min of lessons, 30 min of practice, 35 min for the quiz

Get the most out of it

Lessons

Each lesson has interactive figures, worked examples and questions with instant feedback.

Practice, tools and reference

Use these alongside the lessons, or on their own when you revise.

Requirements

Any modern browser. Works offline.

  • A current version of Chrome, Edge, Firefox or Safari, with JavaScript on. A laptop or desktop screen works best; the pages also fit tablets and phones.
  • No internet connection, installation or account. Everything is inside this folder.
  • Your progress is saved in this browser on this device. Private or incognito windows do not keep it.

Something not working? Read README.txt in the module folder for how to open the module and fix common problems.