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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

Î”x(t)=v0t+at2/2

Q: How do you derive the equations of motion given constant acceleration using calculus?

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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.

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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.

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If an object's acceleration is zero at a specific instant in time, its velocity can either be zero or a constant non-zero value at that instant. This means that the object could be either at rest or moving with a constant velocity at that particular moment.

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The formula for centripetal acceleration can be derived from the equation for centripetal force, F = mω²r. By rearranging this equation to solve for acceleration (F = ma), we get a = ω²r, which represents the centripetal acceleration formula.ω is the angular velocity in radians per second, r is the radius of the circular path, and a is the centripetal acceleration.

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.

To derive the mean of generalized Pareto distribution you must be good with numbers. You must be good in Calculus, Algebra and Statistics.

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

Richard C. Weimer has written: 'Applied calculus with technology' -- subject(s): Calculus, Computer-assisted instruction, Derive, TI-92 (Calculator)

Considering Maxwell equations and contitutive relations. See pag.18 of principles of nano-optics, Lucas Novotny.

Acceleration is the rate of change of velocity over time. By dividing a unit of velocity by a unit of time, we can derive the unit of acceleration. For example, if velocity is measured in meters per second (m/s) and time is measured in seconds (s), acceleration would be in meters per second squared (m/s^2).

Well it is used in certain Chemical equations and to derive other constants. As an example the charge on an electron = 1.6019 x 10-19 coulombs. So a mole of electrons will be 6.023 x 1023 (Avagadro) x 1.6019 x 10-19 coulombs per mole = 96495 coulombs per mole which is Faraday's constant

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