Impulse & Momentum

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Discussion

Momentum keeps me going.

From the Principia.

Quantitas motus est mensura ejusdem orta ex velocitate et quantite materiæ conjunctim.   The quantity of motion is the measure of the same, arising from the velocity and the quantity of matter conjunctly.

We now call this quantity momentum. Momentum is resistance to stopping. It's a kind of inertia for moving bodies.

p = mv

Units

[kg·m/s]

More form the Principia.

Mutationem motus proportionalem esse vi motrici impressæ, & fieri secundum lineaum rectam qua vis illa imprimitur.   The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

Force is directly proportional to the rate of change of momentum with time.

F = m a

F = m Δv/Δt

FΔt = m Δv

The quantity on the right is the change in momentum (Δp = mΔv). That should be easily seen. The quantity on the right is something new. We'll call it impulse and represent it with the letter J. (I've also seen the more sensible letter I used from time to time.) Thus …

J = F Δt

or in the language of calculus

J =  ⌠
⌡		 F dt

Units

[N·s]

This relationship is called the impulse-momentum theorem. In words "impulse causes a change in momentum".

J = Δp

Maybe because the use of the letter "J" to represent a quantity whose name begins with the letter "I" is so odd, this relationship is usually written in its expanded form.

F Δt = m Δv

In a way, this is a nice convention since now we can see the equivalence of units a bit more easily.

[N·s = kg·m/s]

Summary

Problems

practice

  1. Which would you rather be hit by: the fastest pitched baseball or the fastest served tennis ball? To answer this question determine the following quantities …
    1. the momentum of each ball
    2. the impulse delivered if the baseball stops dead on impact but the tennisball rebounds at nearly the same speed.
    Write your answers in the table below.

    quantity baseball tennisball
    mass 145 g 57 g
    speed 45 m/s 75 m/s
    momentum
    impulse

    • Answer it.
  2. Analyze This:
    Now, the bears I live with average, the males, eight to twelve hundred pounds [360-540 kg]. They're the largest bears in the world …. They've been clocked at 41 [mph] and they've run a hundred meter dash in 5.85 seconds, which a human on steroids doesn't even approach.
    Timothy Treadwell author of Among Grizzlies. The Late Show with David Letterman. NBC. 20 February 2001.
    • Answer it.
  3. Analyze This:
    about one atom per cubic centimeter … At two-tenths the speed of light, dust and atoms might not do significant damage even in a voyage of 40 years, but the faster you go, the worse it is -- space begins to become abrasive. When you begin to approach the speed of light, hydrogen atoms become cosmic-ray particles, and they will fry the crew …. So 60,000 kilometers per second may be the practical speed limit for space travel.
    Isaac Asimov. "Sail On! Sail On!" The Relativity of Wrong. New York: Kensington, 1996: 220.
    • Answer it.
  4. Write something completely different.
    • Answer it.

conceptual

  1. In older passenger cars, body panels were attached to a single frame around the perimeter, making them very rigid. This is known as body-over-frame construction. In newer cars, different body parts have stress-bearing elements within them and these parts are then welded to each other. This is known as unitized body construction. Repairing "unibody" cars after collision is comparatively difficult as stress (and thus damage) are distributed throughout the different parts. Why then are cars now built this way?
  2. To escape from a horrible fire, two people are forced to jump from the third story of a burning building on to solid concrete. Which person is more likely to sustain serious injuries: the jumper who comes to an abrupt halt when he lands or the jumper who bounces after impact?

numerical

  1. When hit, the velocity of a 0.145 kg baseball changes from +20 m/s to −20 m/s. What is the magnitude of the impulse delivered by the bat to the ball?
  2. A rubber ball of mass 0.025 kg traveling at 4.0 m/s down strikes the floor and bounces straight up at 2.0 m/s. Find the magnitude of the impulse that the floor gave to the ball.
Plate
Antarctic
African
Arabian
Caribbean
Cocos
Eurasian
Indian
Nazca
North American
Pacific
Philippine
South American

investigative

  1. Determine the momentum of one of earth's tectonic plates (sometimes referred to as continental plates). A list of the most popular plates is shown on the right.
    1. Answer these three related questions first.
      1. Which plate have you chosen to work with?
      2. What is the speed of this plate?
      3. What is the area of this plate?
    2. Now answer these three related questions.
      1. Is the plate you've chosen continental or oceanic?
      2. What is a typical density of this kind of crust?
      3. What is a typical thickness of this type of crust?
    3. Compute the momentum of the tectonic plate you've chosen from the data you've found. State your answer to the nearest order of magnitude (the nearest power of ten). Don't forget the unit.

Resources

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