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Using Eqs. (2.17) and (2.18), derive the expressions for velocity Vx and position x as functions of
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Jan 10, 2025
Using Eqs. (2.17) and (2.18), derive the expressions for velocity Vx and position x as functions of time when the acceleration ax is constant.
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using equations 2.17 and
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2.18 Drive the expression for velocity
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and position velocity P of X and
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position X as function of time when the
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acceleration a of X is constant we are
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to use two equations 2.17 and
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2.18 and the given equations which is
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2.17 is p of xal to B of X Plus integral
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of a of X DT an equation 2.18 is X = x
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plus b of X integral of B of X DT which
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is a function of position and number one
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stands for velocity while X stands for
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position so now we'll look at the
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problem by solving the velocity B of X
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first using the equation 2.17 we will
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have B of xal to B of X integral of a of
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X DT and since a of x from the above
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equation is constant it can be taken
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outside the integral and we end up
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having B of xal B X plus a x of T and
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the reason why is because if you take a
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of T outside a x outside and T integral
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of DT is T now we solve for position X
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substitute substitute B of x isal to B
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of x + a x of T into equation
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2.18 and xal to X Plus integral of when
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we substitute we will have instead of V
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of X we will now have B of X+ X of
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T inside bracket DT expand the integral
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after expanding the integral we have X
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Plus integral of V of X DT plus integral
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of a x of T DT and since B of X is
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constant then we have integral of V of
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xal to
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B integral of V of X DT is equal to B of
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x t why because if you take B not of X
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outside an integral of DT is going to
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give you T therefore you multiply it by
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V and you will have B of x
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t now for integral of a x of T DT we
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will apply the same logic and say
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integral of a x of T DT you equal
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to take a x outside and then integral of
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T DT will give you
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t² / 2 therefore we have half a x
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t² which is the which is the second
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equation therefore we combine the terms
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and get X = X plus b of X plus b a x of
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t² therefore the results will now be the
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velocity as function of time which is B
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of x equal B not of x + a x of
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T the two for position as function of
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time will now have x = x + B of x * t +
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half ax of
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t² so as you can see from equation from
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the first
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