1.4: Kinematics
Equations of Motion
Okay, enough of the definitions. Let’s see how these things all fit together, and how they can be used. What we will be looking at are called the equations of motion , and this topic is often referred to as kinematics . It is important to note that we are not yet dealing with causes for these motions, but only the motions themselves.
We will mostly only deal with constant accelerations (unless otherwise specified), and since instantaneous acceleration is the derivative of velocity, it is not difficult to integrate it to get the instantaneous velocity as a function of time:
The constant of integration is found by plugging \(t = 0\) into Equation 1.4.1 , which results in the velocity of the object at the starting time, which is typically designated as \(v_o\).
We can play exactly the same game to obtain the equation of motion for position as a function of time, since we know how it relates to the instantaneous velocity:
\[\left. \begin{array}{l} v = \dfrac{dx}{dt}\;\;\; \Rightarrow \;\;\;x\left( t \right) = \int {vdt} = \int {\left( {at + {v_o}} \right)dt} = \frac{1}{2}a{t^2} + {v_o}t + const \;\\ const = x\left( {t = 0} \right) \equiv {x_o} \end{array} \right\}\;\;\;x\left( t \right) = \frac{1}{2}a{t^2} + {v_o}t + {x_o}\]
Notice that if we have all the details of this last equation, we can obtain the velocity equation above simply by taking a derivative. We cannot go in the opposite direction without also obtaining the starting position.
Analyze This
A particle moves along the \(x\)-axis with an acceleration that varies linearly with time.
- Analysis
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First, we note that this is not a case of constant acceleration, so equations 1.4.1 and 1.4.2 do not apply. But the calculus we employed to get to these equations does apply, so we just need some sort of mathematical expression for acceleration to repeat that process. We are given that the acceleration varies linearly with time, so translating this into a mathematical expression gives:
\[a\left(t\right)=\lambda t + \beta\nonumber\]
where \(\lambda\) and \(\beta\) are unknown constants. Note that the acceleration at time \(t=0\) is just \(\beta\), so it is more descriptive to just call it \(a_o\) from this point on.
Without these constants (and some others), we cannot compute values like speeds and positions at different times, but we can still do some calculus in terms of the unknowns. The velocity of the particle at a time \(t\) in terms of the acceleration is:
\[v\left(t\right)=\int a\left(t\right) dt = \int \left(\lambda t + a_o\right) dt = \frac{1}{2}\lambda t^2 + a_o t + \gamma\nonumber\]
where \(\gamma\) is the constant of integration. We note that at \(t=0\) the velocity just equals \(\gamma\), so hereafter we'll just call that constant \(v_o\).
We can repeat this process for the position of the particle as a function of time (noting that the constant of integration this time is the position at time \(t=0\)):
\[x\left(t\right)=\int v\left(t\right) dt = \int \left(\frac{1}{2}\lambda t^2 + a_o t + v_o\right) dt = \frac{1}{6}\lambda t^3 + \frac{1}{2}a_o t^2 + v_o t + x_o\nonumber\]
Let’s make an accounting of all the numbers we can encounter in a constant-acceleration situation:
- independent variable: \(t\)
- dependent variables: \(x\) , \(v\)
- constants of the motion: \(x_o\) , \(v_o\) , \(a\) (acceleration is constant by assumption)
With six numbers to work with, you can imagine there are many ways to set up a problem to solve for something unknown. Everything you need to solve any such problem is provided in the above equations. However, it is often easier to put those equations together to form a new equation, to cut down on the algebra needs for certain types of problems. The most common useful re-combining of these variables involves eliminating time from the two equations, since you may be given velocities and positions. The algebra is straightforward:
You can think of this equation as the “before/after” equation, because it deals only with starting and ending positions and velocities, and has eliminated time as an input variable.
While we are accumulating useful (though unnecessary) equations for motion with constant acceleration, we should also include the two equations that involve average velocity. The first is just a rewriting of the definition of average velocity, with the "final" position occurring at time \(t\):
\[{v_{ave}} = \dfrac{x_f - x_o}{t} = \dfrac{x \left( t \right) - x_o}{t}\;\;\;\Rightarrow \;\;\; x \left( t \right) = v_{ave}t + x_o\]
The second equation is quite useful, though it applies only to motion involving constant acceleration:
\[{v_{ave}} = \dfrac{x_f - x_o}{t} = \dfrac{\frac{1}{2}a{t^2} + {v_o}t}{t} = \frac{1}{2}at + v_o = \frac{1}{2}\left( v_f - v_o \right) + v_o \;\;\; \Rightarrow \;\;\; v_{ave} = \dfrac{v_o + v_f}{2} \]
For constant acceleration, the average velocity simply equals the arithmetic average of the starting and ending velocities. We will better see why it comes out this way when we start discussing graphing shortly.
Free-Fall
There is one type of straight-line motion that involves constant acceleration that we are all familiar with: free-fall.
We will look more closely at how to explain this in terms of forces in a future section, but assuming air resistance has a small effect (remember, we are devising a simplified model here), then it turns out (as shown by Galileo dropping stones from the Tower of Pisa, and more dramatically in the demonstration) that objects all accelerate at the same constant rate as they fall to Earth. This rate of acceleration is commonly given the symbol \(g\), and it has the value:
acceleration due to gravity near the surface of the earth \( = g = 9.8\frac{m}{s^2}\)
Note the units of distance-per-time-squared are the units of acceleration. This acceleration is of course always directed downward, and depending on our choice of coordinate system, this can be either positive or negative. Once the coordinate system is selected, the sign for \(g\) stays the same no matter which way the object is moving. If the positive direction is chosen to be upward, and the object is moving upward, then its velocity is positive and the negative value of \(g\) leads to a slowing of the object’s motion. If it is moving down, then its velocity is negative, and the negative acceleration leads to the velocity becoming more negative (i.e. it is speeding up).
A ball is thrown vertically upward at the same instant that a second ball is dropped from rest directly above it.
- Analysis
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Both balls are under the influence of the earth's gravity, and therefore both accelerate at a rate \(g\) downward. However, one starts with a velocity in the upward direction, while the other starts from rest. With both balls subject to the same velocity equation:
\[v\left(t\right)=-gt+v_o\nonumber\]
their different values for \(v_o\) ensure that they will always have different velocities. In particular, when the ball thrown upward is eventually moving downward, then assuming there is no collision, its speed will always be less than the speed of the other ball. This means that if the two balls start sufficiently high above the Earth's surface, they are guaranteed to eventually collide.
Suppose the higher ball starts at a distance of \(y_o\) above the lower ball. Calling the initial speed of the lower ball \(v_o\), and calling its starting height zero, we can write down the equations of motion of both balls:
\[\begin{align} upper: && y_{higher}\left(t\right)=-\frac{1}{2}gt^2+y_o \nonumber\\ lower: && y_{lower}\left(t\right)=-\frac{1}{2}gt^2+v_o t \nonumber\end{align}\nonumber\]
Naturally, if we are interested in when the two balls collide, we simply set the two heights equal to each other. If we do this, this seemingly complicated situation reduces to something very simple, as the effects of the gravitational accelerations of the two balls cancel out, which means that they will collide when the lower ball traverses the separation with its initial velocity, ignoring the common acceleration:
\[y_{higher}=y_{lower}\;\;\;\Rightarrow\;\;\; v_o t_{collision} = y_o\nonumber\]


11.5: Factors Affecting Soil Development
Soil research has shown that soil profiles are influenced by five separate, yet interacting, factors: parent material, climate, topography, organisms, and time. Soil scientists call these the factors of soil formation . These factors give soil profiles their distinctive character.
Parent Material
Soil parent material is the material that soil develops from, and may be rock that has decomposed in place, or material that has been deposited by wind, water, or ice. The character and chemical composition of the parent material plays an important role in determining soil properties, especially during the early stages of development.
Soils developed on parent material that is coarse grained and composed of minerals resistant to weathering are likely to exhibit coarse grain texture. Fine grain soil develop where the parent material is composed of unstable minerals that readily weather.
Parent material composition has a direct impact on soil chemistry and fertility. Parent materials rich in soluble ions-calcium, magnesium, potassium, and sodium, are easily dissolved in water and made available to plants. Limestone and basaltic lava both have a high content of soluble bases and produce fertile soil in humid climates. If parent materials are low in soluble ions, water moving through the soil removes the bases and substitutes them with hydrogen ions making the soil acidic and unsuitable for agriculture. Soils developed over sandstone are low in soluble bases and coarse in texture which facilitates leaching. Parent material influence on soil properties tends to decrease with time as it is altered and climate becomes more important.
Climate
Soils tend to show a strong geographical correlation with climate, especially at the global scale. Energy and precipitation strongly influence physical and chemical reactions on parent material. Climate also determines vegetation cover which in turn influences soil development. Precipitation also affects horizon development factors like the translocation of dissolved ions through the soil. As time passes, climate tends to be a prime influence on soil properties while the influence of parent material is less.
Climate, vegetation, and weathering
Climate affects both vegetative production and the activity of organisms. Hot, dry desert regions have sparse vegetation and hence limited organic material available for the soil. The lack of precipitation inhibits chemical weathering leading to coarse textured soil in arid regions. Bacterial activity is limited by the cold temperatures in the tundra causing organic matter to build up. In the warm and wet tropics, bacterial activity proceeds at a rapid rate, thoroughly decomposing leaf litter. Under the lush tropical forest vegetation, available nutrients are rapidly taken back up by the trees. The high annual precipitation also flushes some organic material from the soil. These factors combine to create soils lacking much organic matter in their upper horizons.
Climate, interacting with vegetation, also affects soil chemistry. Pine forests tend to dominate cool, humid climates. Decomposing pine needles in the presence of water creates a weak acid that strips soluble bases from the soil leaving it in an acidic state. Additionally, pine trees have low nutrient demands so few soil nutrients are taken back up by the trees to be later recycled by decaying needle litter. Broadleaf deciduous trees like oak and maple have higher nutrient demand and thus continually recycle soil nutrients keeping soils high in soluble bases.
Topography
Topography has a significant impact on soil formation as it determines runoff of water, and its orientation affects microclimate which in turn affects vegetation. For soil to form, the parent material needs to lie relatively undisturbed so soil horizon processes can proceed. Water moving across the surface strips parent material away impeding soil development. Water erosion is more effective on steeper, unvegetated slopes.
Effect on soil erosion
Slope angle and length affects runoff generated when rain falls to the surface. Examine the diagram below showing the relationship between hill slope position, runoff, and erosion.
The amount of water on a particular hill slope segment is dependent on what falls from precipitation and what runs into it from an upslope hill slope segment. The hill slope in Figure \(\PageIndex{2}\) has been divided into several segments and the amount of precipitation falling on each segment is the same. As water runs down slope, the water that has accumulated in segment A runs off adding to what falls into segment B by precipitation. The water in B runs into C, and C into D, and so on. The amount of water increases in the down slope direction as water is contributed of water from upslope segments. The velocity of the water increases as well as it moves towards the base of the slope. As a result, the amount and velocity of water, and hence rate of erosion increases as you near the base of the slope. Rather than infiltrating into the soil to promote weathering and soil development, water runs off. Erosion causes stripping of the soil thus preventing parent material to stay in place to develop into a soil. So we should expect to find weakly developed soil at the mid- and near the bottom of the slope.
Effect on deposition and soil texture
Water velocity not only determines the rate of erosion but the deposition of soil material in suspension too. Figure \(\PageIndex{3}\) shows the relationship between location and texture. Sites A, B, and C, are located progressively further from the base of a slope. A soil texture triangle is used to illustrate the variation in soil textures at the three sites.
As water empties from a mountain stream, its velocity starts to decrease. The largest size particles, like sand, are the first to drop out of suspension (Site A). Fine, clay size particles can be carried further away from the base of the slope before they are deposited. As a result, coarse textured soils tend to be found near the base of the mountain and fine textured soils are located further away (Site C).
Microclimatic effects
Hill slope orientation affects the microclimate of a place. As the slope of the surface increases, so does the local sun angle, up to a point. As the local sun angle increases, the intensity of heating increases, causing warmer surface temperatures and, likely, increased evaporation. Orientation of the hill slope is certainly important too. Those slopes which face into the sun receive more insolation than those facing away. Thus inclined surfaces facing into the sun tend to be warmer and drier, than flatter surfaces facing way from the sun. The microclimate also impact vegetation type.
Organisms
Organism, both plant and animal, play an important role in the development and composition of soil. Organisms add organic matter, aid decomposition, weathering and nutrient cycling. The richness and diversity of soil organisms and plant life that grows on the surface is, of course, also tied to climate.
Nutrient cycling
Biotic elements of the environment need life-sustaining nutrients that find their origin in the soil. Upon their death, organisms return these nutrients to the soil to be taken up again by other plants and animals. Hence there is a constant cycling of nutrients between organisms and soils. This cycling refreshes and maintains the nutrient status of soils. Without it, soluble nutrients would be leached from the soil, decreasing the soil's ability to support life.
The degree to which nutrients are cycled depends on the needs of the organism occupying a particular place. For instance, broadleaf, deciduous trees like oak and maple generally have high nutrient demand creating surface litter rich in nutrients when leaves die and fall to the forest floor. Decomposition of the litter releases the nutrients back into the soil for the tree to take back up. Thus soils under these kinds of forests tend to be high in soluble bases and nutrients.
Pine trees generally have low nutrient demands. The decaying litter that falls to the forest floor is poor in nutrients. As a result, little cycling of soluble nutrients like calcium, magnesium, sodium, and potassium occurs and are thus leached creating an acidic soil environment.
Organisms and weathering
Soil organisms also affect weathering. The decomposition of pine needles creates a weak acid that can strip soluble ions from the soil. Burrowing animals create passage ways through the soil to help aerate and allow water to infiltrate into it. Burrowing animals help translocate materials and fertilize the soil at depth.
Time
As time passes, the weathering processes continue to act on soil parent material to break it down and decompose it. Horizon development processes continue to differentiate layers in the soil profile by their physical and chemical properties. As a result, older more mature soils have well-developed sequence of horizons, though some may undergo so much weathering and leaching that visually distinct layers may be hard to see. This is a notable characteristic of oxisols. Some geological processes keep soils from developing by constantly altering the surface and thus not allowing parent material to weather over a significant period of time. For instance, erosion of hillsides constantly removes material thus impeding soil development. Along the channels of rivers, new sediment is frequently deposited as the river spills out onto its floodplain during floods. The constant addition of new material restarts the soil development process.
Climate interacts with time during the soil development process. Soil development proceeds much more rapidly in warm and wet climates thus reaching a mature status sooner. In cold climates, weathering is impeded and soil development takes much longer.
Video : Watch "The Five Factors of Soil Formation" (Courtesy of UBC Virtual Soil Learning Resources)