Comments due by midnight July 11th
Use and cite the reading to support your responses.
What does the slope of two points on velocity time graph represent?
How is this different than the average velocity?
How is it different than the instantaneous velocity?
What are two other ways to describe the above?
Head Banger question:
Can I have a velocity if no time has passed? explain your reasoning.
-A limit falls under the "limitating process" (Page 3) This process is "there is some true value" of an instantaneous velocity that approaches "this value for suffiently small values of" change in time. Limits are one of the foundations in calculus and is arguably the biggest concept discussed writhin the topic. It is the subject that formed derivatives, integrals, etc.
ReplyDelete-The slope of two points from a velocity-time graph represents acceleration. This type of acceleration is known as average acceleration since it is found from two points from a velocity and time graph. The formula for such acceleration would be acceleration = (final velocity - initial velocity) / time. However, the instantaneous acceleration definition would be represented as acceleration = dv(t)/dt, where dv(t)/dt is the "limiting process. This information can be found on pages 6 and 7.
-Acceleration is different from average velocity because average velocity would be slope of two points from a POSITION and time graph. Average acceleration is the slope of two points on a VELOCITY and time graph.
-Average velocity is different then instantaneous velocity because in simple terms, average velocity is the slope of two points from a position and time graph (secant) while instantaneous velocity is the slope of one point of a position and time graph (tangent) thus making the velocity appearing at one location "instantly." Even though velocity is the change of position over time, the instantaneous velocity undergoes a limitation process velocity = limit of change in R divided change in t as change in t approaches 0. For the actual derivative function, velocity = dR(t)/dt. "The instantaneous velocity vi at position i is the limit, as delta t goes to zero, if the ratio delta R/delta t" (page 5).
-One way to describe instantaneous velocity is the derivative of position at a point in time. Remember that the proven theorem of v = dR(t)/dt makes the "instantaneous velocity" the derivative of the position from the limiting process.
Another way to name the instantaneous velocity is by describing it as the integral of acceleration at a certain point in time. Again, acceleration is the "slope" or in calculus, the derivative (as long as it is at only one point in time). The formula a= dv(t)/dt shows that the formula that ∫a dt = vt. This means that the integral of acceleration is velocity. Information can be found from pages 4 to 7.
-Velocity cannot exist without time. Time is the basis of everything and in physics, it's the variable. Without a variable, it would represent nothing. So if velocity is the limit of change in position over the change in time as the change in time approaches zero, two complications would occur:
1. Time would be undefined so velocity would be undefined
2. A variable cannot be nothing and then approach a value thus making the value and velocity irrelevant.
1. A limit is based on the “limiting process” (2). The basis of this process is that there is an actual value, “called the instantaneous velocity”, that is approached “for sufficiently small values of Δt.” As Brandon said, limits are fundamental to calculus, as they are used to find derivatives, integrals, etc. All of calculus branches off of limits; they are essential to the subject.
ReplyDelete2. The slope of two points on a velocity vs time graph represents average acceleration, shown by the formula a= (vf-vi)/Δt (7).
3. Average acceleration is different than the average velocity because it is analyzed on a velocity vs time graph instead of a position vs time graph. Velocity is the derivative of a position function, therefore the slope on a position vs time graph would give average velocity (5).
4. Average velocity differs from instantaneous velocity because the average velocity is the slope of a position vs time graph (over a certain period of time). Instantaneous velocity, however, is at a particular instant and utilizes the limitation process, shown by instantaneous velocity= the limit at time approaches 0 of the change in position over change in time (6), or more simply put, the derivative of the position function at time t.
5. As aforementioned, instantaneous velocity is the position function’s derivative. As seen on page 6, the “streamlined” notation of the limit definition of instantaneous velocity is in fact the derivative of the position function. A second way to describe instantaneous velocity is acceleration’s integral. Because the derivative of velocity is acceleration, the integral, or antiderivative, is velocity. The integral of acceleration, the “displacement” (9) of velocity vectors, given specific bounds, will evaluate the velocity at a particular instant.
**Head Banger
One cannot have a velocity if no time has passed. Velocity is calculated by the change in position over the change in time. If there is no time--no value for this necessary variable-- velocity cannot be determined. Time is essential in all of physics and calculus; it is a crucial variable in all data. Not only velocity, but acceleration as well could not be identified without time. Velocity and acceleration, along with all related functions, are dependent on time. If no time passes, the variable serves no purpose, and no data could be collected.
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ReplyDelete1. A limit, as shown in figure 1, is a result of time intervals getting infinitely small until it is essentially zero. Known as the "limiting process" (p. 1), this ends in the ability to determine an instantaneous velocity, as opposed to an average velocity over an interval of time. This is the basis to much of calculus, including derivatives and integrals.
ReplyDelete2. The slope between two points on a velocity graph represents average acceleration. Much like how the slope on a position graph represents velocity, the same can be said for the "next step up." As shown on p. 6, a(t)=dv(t)/dt.
3. Like I said in the previous answer, acceleration is the "next step up." The average velocity would be found as the slope between two points on a position graph, whereas average acceleration is the slope between two points on a velocity graph.
4. Comparing average velocity vs. instantaneous velocity, average velocity is the slope between two points, or a time interval, on a position graph while instantaneous velocity is the derivative (limit) at a specific, singular time t on a position graph. This is explicitly sated on p. 4, "the instantaneous velocity vi at position i is the limit, as Δt goes to zero, or the ration ΔRi/Δt."
5. Like mentioned before, instantaneous velocity can be described as the limit as Δt goes to zero of the ratio ΔRi/Δt, or also can be notated as v(t)=dR(t)/dt, signifying a derivative. All of this is written out on pp. 4-5.
Head Banger: If we are saying that no time has passed, wouldn't that be the same as saying Δt is approaching 0? Having an infinitely small Δt is the same as having no Δt at all. The limit as Δt approaches 0 is the basis of instantaneous velocity, and we know very well that instantaneous velocity exists. Therefore I believe that velocity with a Δt of 0 does exists, in the form of instantaneous velocity.
1. A limit is based on the limiting process(pg. 1). Figure 1 shows how the limiting process involves a transition in to instantaneous velocity. The time interval is getting infinitely smaller until it is basically zero. The limiting process allows one to find an instantaneous velocity.
ReplyDelete2. The slope between two points on a velocity graph represents average acceleration. a(t)=dv(t)/dt (pg. 6)
3. Average velocity would represent the slope between two points on a position graph. On the other hand, average acceleration represents the slope between two points on a velocity graph.
4. Instantaneous velocity involves using the limiting process and derivatives at a specific time t on a position graph. "The instantaneous velocity vi at position i is the limit, as Δt goes to zero, or the ration ΔRi/Δt." (pg. 4) Average velocity is a measure of the distance traveled in a given period of time, or the slope between two points on a position graph.
5. Instantaneous velocity can be described as the derivative of the position graph at a specific time t. Another way to describe instantaneous velocity is the integral of an acceleration graph. The derivative of velocity is the acceleration, so the integral can be used when given acceleration to go back to velocity.
HEADBANGER.
If no time has passed, the position graph will just hold a singular point at time 0. There is no way, that I know of, to be able to tell which direction or how quickly the object would be moving. No matter how small the smallest amount of time that passes, information can still be used to determine the velocity of the object. Also, since velocity is measured by the change in position over the change in time, you cannot divide by 0 which would represent the change in time. Without time, there isn't enough information to determine an objects velocity.
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ReplyDeleteThis comment has been removed by the author.
ReplyDelete1. A limit can be defined as a value that a dependent variable/value is approaching as the independent variable is approaching a specified value. (p.1 and 5)
ReplyDelete2. The slope represents the average acceleration from the first point to the second point as acceleration is defined as the change in velocity over the change in time. (p. 7)
3. Average velocity differs from average acceleration as average velocity is defined as the displacement over the change in time while acceleration is the change in velocity over the change in time. This means average velocity is the integral of average acceleration, and average velocity would be expressed as the slope between two points on a distance graph instead of a velocity graph. (p.6,7, and 8)
4. Instantaneous velocity differs from average velocity as average velocity expresses the mean velocity over a defined period of time while instantaneous velocity is the velocity at an exact moment and thus it applies to an undefined period of time which is infinitely small. This means that while average velocity can be calculated exactly, instantaneous velocity must be estimated with limits as the time is undefined. (p. 5 and 7)
5. Instantaneous velocity can be described in many different ways. As referenced before instantaneous velocity can be defined as the limit value of the velocity as the change in time approaches zero. Instantaneous velocity can also be described as the integral of acceleration as instantaneous velocity is the value of the area under the acceleration graph added to the initial velocity.
(p.5, 14, and 20)
Head Banger: Velocity cannot exist if no time has passed. Velocity is defined as displacement over the change in time. As the change in time is the denominator, when it is zero, the value of the velocity is undefined, and thus velocity is nonexistent. While instantaneous velocity can be used as a theoretical value that is estimated using limits, in the real world, velocity clearly cannot exist without time passing. (p. 5 and 7)
1. A limit is the value that some function approaches as its variable approaches a certain value. For example, instantaneous velocity is the limit of the an object's velocity as Δt approaches 0(p.1).
ReplyDelete2.The slope of two points on a velocity graph represents the average acceleration during the time between the two points. Slope is rise over run on a graph; in this case, the rise is the change in velocity, and the run is the change in time. This gives you Δv/Δt, which is defined as acceleration. (p.6)
3. This is different to average velocity because average velocity is the slope between two points on a position time graph, while average acceleration is the slope between two points on a velocity time graph. Essentially, average acceleration is the derivative of velocity. (pp.5,6)
4. Instantaneous velocity is different from average acceleration for the same differences between velocity and acceleration as stated in my answer to question 3, but is also different because the differences between "instantaneous" and "average." In anything "average", such as average acceleration, Δt must be, no matter how small, finite. On the other hand, instantaneous velocity is the limit of the same expression (average velocity in this case), as Δt approaches 0. (pp.4,6)
5. Instantaneous velocity can be described as the derivative of a position time graph at a certain t. It can also be the the integral of the acceleration plus the initial velocity.(p. 19)
Head Banger: There can not be velocity if there has been no passage of time. Velocity is defined as ΔR/Δt, so if no time has passed (Δt =0), then you would be dividing by zero. In mathematics, this point would be a singularity.
Well done☺ I would expect nothing less. If you have not responded or participated you have a 0 on module 1.
ReplyDelete1. A limit is a value that is found as a function approaches another, preset value. (p.1)
ReplyDelete2. The two points on a velocity-time graph represent the acceleration during a given time. (cal 1-4)
3. Acceleration is different to average velocity because acceleration is the slope on the v-t graph while the average velocity is the midpoint of the line between two points on the v-t graph. (cal 1-5)
4. Acceleration is not instantaneous velocity because acceleration must happen over a period of time. Instantaneous velocity is the velocity of an object at a given instance, meaning no time can pass. “the instantaneous velocity
vi at position i is the limit, as Δt goes to zero, of
the ratio ΔR i /Δt . ” (cal 1-5)
5. Instantaneous velocity can be calculated in two ways. One way is to find the derivative of the position equation during a set period. The other way is to integrate the acceleration equation during a set period. Instantaneous velocity can be described as both the derivative of the position and the integration of the acceleration. (cal 1-8)
HB. Velocity cannot exist if time does not exist because the definition of velocity is the change in position over time. However, if time is merely stopped, objects can retain an instantaneous velocity.