To derive the kinematic equations of motion in one dimension with a given acceration 'a(t)', one begins with the definition of acceleration: the change in velocity per unit time.
average acceleration = the change in velocity/time elapsed
Acceleration, technically instantaneous acceleration, is the average acceleration over a very small interval of the velocity/time function. Instantaneous acceleration (hereafter referred to simply as 'acceleration' or 'a') is then, by extension
a = limitt-->0(instantaneous velocity1 - instantaneous velocity2)/t
which is the definition of the derivitive of instantaneous velocity ('v') with respect to time ('t'). Thus we have:
a= dv/dt
because velocity is itself change in position ('x') we can similarly derive
v= dx/dt
and
a= d2x/dt2
By the fundamental theorem of calculus:
v= integral(a)dt +C
x=integral(v)dt +C
in order to eliminate the arbitrary constant C, we use initial conditions:
v0=v(0), a0=a(0), etc.
any function representing the motion of real quantities according to the principles of classical mechanics has the value 0 for all integrals taken from an arbitrary point b to the same point b, where b is within its domain. Thus:
v(0)= 0 +C
v0=C
and so for all of the other quantities. Thus we yield:
v= v0 + integral(a)dt
x= x0 + integral(v)dt
in the special case of constant acceleration, we can take those integrals:
integral(a)dt= at
integral(v)dt= integral(v0+at)= v0t + at2/2
so our final formulae are:
v(t)=v0+at
Δx(t)=v0t+at2/2
Independence:The equations of a linear system are independent if none of the equations can be derived algebraically from the others. When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set.Consistency:The equations of a linear system are consistent if they possess a common solution, and inconsistent otherwise. When the equations are inconsistent, it is possible to derive a contradiction from the equations, such as the statement that 0 = 1.Homogeneous:If the linear equations in a given system have a value of zero for all of their constant terms, the system is homogeneous.If one or more of the system's constant terms aren't zero, then the system is nonhomogeneous.
Considering Maxwell equations and contitutive relations. See pag.18 of principles of nano-optics, Lucas Novotny.
barn
It is not possible to reproduce the equations on this website, however you can find a detailed derivation at the related link.
If you're able to get around in Calculus, then that derivation is a nice exercise in triple integration with polar coordinates. If not, then you just have to accept the formula after others have derived it. Actually, the formula was known before calculus was invented/discovered. Archimedes used the method of exhaustion to find the formula.
You can't derive the velocity from the acceleration. Zero acceleration simply means that the velocity (at that instant) is not changing.
That is done via calculus. Specifically, take the movement over a small distance, calculate the change in velocity divided by the time, and figure out what happens if the time interval gets smaller and smaller - as they say in calculus, "get the limit of the acceleration as the time tends towards zero".
rmsvoltage
Independence:The equations of a linear system are independentif none of the equations can be derived algebraically from the others. When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set.Consistency:The equations of a linear system are consistent if they possess a common solution, and inconsistent otherwise. When the equations are inconsistent, it is possible to derive a contradiction from the equations, such as the statement that 0 = 1.Homogeneous:If the linear equations in a given system have a value of zero for all of their constant terms, the system is homogeneous.If one or more of the system's constant terms aren't zero, then the system is nonhomogeneous.
Independence:The equations of a linear system are independent if none of the equations can be derived algebraically from the others. When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set.Consistency:The equations of a linear system are consistent if they possess a common solution, and inconsistent otherwise. When the equations are inconsistent, it is possible to derive a contradiction from the equations, such as the statement that 0 = 1.Homogeneous:If the linear equations in a given system have a value of zero for all of their constant terms, the system is homogeneous.If one or more of the system's constant terms aren't zero, then the system is nonhomogeneous.
You can derive it from accelerating an object to a certain speed. Assume constant acceleration (and therefore constant force), and calculate how much work (force x distance) you need to get the object to a specific speed.
To derive the mean of generalized Pareto distribution you must be good with numbers. You must be good in Calculus, Algebra and Statistics.
You have a contradiction in your question. Instantaneous acceleration is the acceleration at a certain moment in time. Average acceleration is the average over a time interval.
F=M(A), you can simply derive a formula by solving for A. So devide F by M and you get A=F/M. Then you can ask yourself, if when you increase of decrease mass what will happen to acceleration. assuming the unbalanced force is constant. soo when mass increases acceleration decreases. and when you take away mass from a body, then you can say that acceleration increases. You must assume that the force is constant. :D
Considering Maxwell equations and contitutive relations. See pag.18 of principles of nano-optics, Lucas Novotny.
Richard C. Weimer has written: 'Applied calculus with technology' -- subject(s): Calculus, Computer-assisted instruction, Derive, TI-92 (Calculator)
1) What is the definition of dielectric permittivity on the basis of Maxwell equations? 2) What is Poisson equation of Electrostatics? Derive the Poisson equation from Maxwell equations. 3) Write the Biot-Savar equation. What is the meaning? 4) Derive a wave equation of a plain electromagnetic wave from Maxwell equation.