#set document(title: "9.1 Linear Momentum", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 9.1#h(0.6em)Linear Momentum Our study of kinetic energy showed that a complete understanding of an object’s motion must include both its mass and its velocity (#math.equation(block: false, alt: "K equals open parenthesis 1 / 2 close parenthesis m v squared")[$K = ( 1 "/" 2 ) m v^(2)$]). However, as powerful as this concept is, it does not include any information about the direction of the moving object’s velocity vector. We’ll now define a physical quantity that includes direction. Like kinetic energy, this quantity includes both mass and velocity; like kinetic energy, it is a way of characterizing the “quantity of motion” of an object. It is given the name #strong[momentum] (from the Latin word #emph[movimentum], meaning “movement”), and it is represented by the symbol #emph[p]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Momentum] The momentum #emph[p] of an object is the product of its mass and its velocity: #math.equation(block: true, alt: "p → equals m v → .")[$arrow(p) = m arrow(v) .$] ] #figure(figph[Photo of a soccer player kicking a ball. Two arrows have been added to the photo at the ball’s position. Both arrows point forward, in the direction the player is kicking. One arrow is labeled velocity, the other arrow is labeled momentum.], alt: "Photo of a soccer player kicking a ball. Two arrows have been added to the photo at the ball’s position. Both arrows point forward, in the direction the player is kicking. One arrow is labeled velocity, the other arrow is labeled momentum.", caption: [The velocity and momentum vectors for the ball are in the same direction.]) As shown in , momentum is a vector quantity (since velocity is). This is one of the things that makes momentum useful and not a duplication of kinetic energy. It is perhaps most useful when determining whether an object’s motion is difficult to change or easy to change over a short time interval. #figure(figph[A photo of a supertanker in the water is shown. There are two much smaller vessels with sails in the distance.], alt: "A photo of a supertanker in the water is shown. There are two much smaller vessels with sails in the distance.", caption: [This supertanker transports a huge mass of oil; as a consequence, it takes a long time for a force to change its (comparatively small) velocity.]) #figure(figph[A drawing of a stoppered flask, labeled “container”, with gas molecules (represented as green dots) moving randomly inside the flask.], alt: "A drawing of a stoppered flask, labeled “container”, with gas molecules (represented as green dots) moving randomly inside the flask.", caption: [Gas molecules can have very large velocities, but these velocities change nearly instantaneously when they collide with the container walls or with each other. This is primarily because their masses are so tiny.]) Unlike kinetic energy, momentum depends equally on an object’s mass and velocity. For example, as you will learn when you study thermodynamics, the average speed of an air molecule at room temperature is approximately 500 m/s, with an average molecular mass of #math.equation(block: false, alt: "6 times 10 to the power −25 kg")[$6 #h(0.2em) × #h(0.2em) 10^(−25) #h(0.2em) "kg"$]; its momentum is thus #math.equation(block: true, alt: "p sub molecule equals open parenthesis 6 times 10 to the power −25 kg close parenthesis open parenthesis 500 the fraction m over s close parenthesis equals 3 times 10 to the power −22 the fraction kg times m over s .")[$p_("molecule") = ( 6 #h(0.2em) × #h(0.2em) 10^(−25) #h(0.2em) "kg" ) ( 500 #h(0.2em) frac("m", "s") ) = 3 #h(0.2em) × #h(0.2em) 10^(−22) #h(0.2em) frac("kg" · "m", "s") .$] For comparison, a typical automobile might have a speed of only 15 m/s, but a mass of 1400 kg, giving it a momentum of #math.equation(block: true, alt: "p sub car equals open parenthesis 1400 kg close parenthesis open parenthesis 15 the fraction m over s close parenthesis equals 21,000 the fraction kg times m over s .")[$p_("car") = ( 1400 #h(0.2em) "kg" ) ( 15 #h(0.2em) frac("m", "s") ) = 21,000 #h(0.2em) frac("kg" · "m", "s") .$] These momenta are different by 27 orders of magnitude, or a factor of a billion billion billion! === Summary - The motion of an object depends on its mass as well as its velocity. Momentum is a concept that describes this. It is a useful and powerful concept, both computationally and theoretically. The SI unit for momentum is kg#math.equation(block: false, alt: "times")[$·$] m/s. === Conceptual Questions An object that has a small mass and an object that has a large mass have the same momentum. Which object has the largest kinetic energy? #solutionbox[ Since #math.equation(block: false, alt: "K equals p squared / 2 m")[$K = p^(2) "/" 2 m$], then if the momentum is fixed, the object with smaller mass has more kinetic energy. ] An object that has a small mass and an object that has a large mass have the same kinetic energy. Which mass has the largest momentum? === Problems An elephant and a hunter are having a confrontation. #figure(figph[A drawing of an elephant, on the left, and hunter, on the right. An x y coordinate system is shown, with positive x to the right and positive y up. The elephant is labeled with m E = 2000.0 k g, and vector v E = 7.50 meters per second times I hat. An arrow above the v E vector points to the right. The hunter is labeled with m hunter = 90.0 k g, and vector v hunter = 7.40 meters per second times I hat. An arrow above the v hunter vector points to the right. Between the hunter and elephant is a dart with a long arrow pointing to the left drawn near it and labeled vector v dart = 600 meters per second times minus I hat, and m dart = 0.0400 k g.], alt: "A drawing of an elephant, on the left, and hunter, on the right. An x y coordinate system is shown, with positive x to the right and positive y up. The elephant is labeled with m E = 2000.0 k g, and vector v E = 7.50 meters per second times I hat. An arrow above the v E vector points to the right. The hunter is labeled with m hunter = 90.0 k g, and vector v hunter = 7.40 meters per second times I hat. An arrow above the v hunter vector points to the right. Between the hunter and elephant is a dart with a long arrow pointing to the left drawn near it and labeled vector v dart = 600 meters per second times minus I hat, and m dart = 0.0400 k g.", caption: none) + Calculate the momentum of the 2000.0-kg elephant charging the hunter at a speed of 7.50 m/s. + Calculate the ratio of the elephant’s momentum to the momentum of a 0.0400-kg tranquilizer dart fired at a speed of 600 m/s. + What is the momentum of the 90.0-kg hunter running at 7.40 m/s after missing the elephant? A skater of mass 40 kg is carrying a box of mass 5 kg. The skater has a speed of 5 m/s with respect to the floor and is gliding without any friction on a smooth surface. + Find the momentum of the box with respect to the floor. + Find the momentum of the box with respect to the floor after she puts the box down on the frictionless skating surface. #solutionbox[ a. magnitude: #math.equation(block: false, alt: "25 kg times m/s;")[$25 #h(0.2em) "kg" · "m/s;"$] b. same as a. ] A car of mass 2000 kg is moving with a constant velocity of 10 m/s due east. What is the momentum of the car? The mass of Earth is #math.equation(block: false, alt: "5.97 times 10 to the power 24 kg")[$5.97 #h(0.2em) × #h(0.2em) 10^(24) #h(0.2em) "kg"$] and its orbital radius is an average of #math.equation(block: false, alt: "1.50 times 10 to the power 11 m")[$1.50 #h(0.2em) × #h(0.2em) 10^(11) #h(0.2em) "m"$]. Calculate the magnitude of its linear momentum at the location in the diagram. #figure(figph[An illustration of the earth orbiting the sun. The mass of the earth is given as 5.97 times 10 to the 24 kilograms and the radius of the orbit is labeled R earth = 1.5 times 10 to the 11 meters.], alt: "An illustration of the earth orbiting the sun. The mass of the earth is given as 5.97 times 10 to the 24 kilograms and the radius of the orbit is labeled R earth = 1.5 times 10 to the 11 meters.", caption: none) #solutionbox[ #math.equation(block: true, alt: "1.78 times 10 to the power 29 kg times m/s")[$1.78 #h(0.2em) × #h(0.2em) 10^(29) #h(0.2em) "kg" · "m/s"$] ] If a rainstorm drops 1 cm of rain over an area of 10 km#super[2] in the period of 1 hour, what is the momentum of the rain that falls in one second? Assume the terminal velocity of a raindrop is 10 m/s. What is the average momentum of an avalanche that moves a 40-cm-thick layer of snow over an area of 100 m by 500 m over a distance of 1 km down a hill in 5.5 s? Assume a density of 350 kg/m#super[3] for the snow. #solutionbox[ #math.equation(block: true, alt: "1.3 times 10 to the power 9 kg times m/s")[$1.3 #h(0.2em) × #h(0.2em) 10^(9) #h(0.2em) "kg" · "m/s"$] ] What is the average momentum of a 70.0-kg sprinter who runs the 100-m dash in 9.65 s?