Question #167028

If 𝑎 = 4𝑡^−3/2 , 𝑠 = 16 𝑤ℎ𝑒𝑛 𝑡 = 4, 𝑎𝑛𝑑 𝑠 = 25 𝑤ℎ𝑒𝑛 𝑡 = 6, find the equation of motion 𝑠 = 𝑓(𝑡) and the velocity function 𝑣(𝑡).


1
Expert's answer
2021-03-01T07:15:40-0500

Acceleration aa is given as


a=dvdt    dv=a  dt    v=a  dta = \dfrac{dv}{dt} \implies dv = a\;dt \implies v = \int a\;dt

The function of the velocity (vv) is given as:


v=4t3/2dt=4t3/2dt=41t32dt=4(2t)v=8t+cv =\int 4t^{-3/2} dt\\ = 4 \int t^{-3/2} dt\\ = 4\int \dfrac{1}{t^\frac{3}{2}} dt\\ = 4 \Bigg(-\dfrac{2}{\sqrt{t}}\Bigg)\\ v=-\dfrac{8}{\sqrt{t}} + c

Velocity (vv) is given as:


v=dsdt    ds=v  dt    s=v  dtv = \dfrac{ds}{dt} \implies ds = v\; dt \implies s = \int v\;dt

The function of the equation of motion can be obtained as follows:


s=(8t+c)dt=8(2t)+cts=16t+ct+c1s= \int \Big(-\dfrac{8}{\sqrt{t}} + c \Big) dt\\ = -8 (2\sqrt{t}) + ct\\ s = - 16\sqrt{t} + ct +c_1

When s = 16 and t = 4:


16=16(4)+c(4)+c1=16(2)+4c+c116=32+4c+c1    4c+c1=48(eqn  1)16 = -16(\sqrt{4})+c(4) + c_1\\ = -16(2) + 4c + c_1\\ 16= -32 +4c +c_1\\ \implies 4c+c_1 = 48 \qquad\dots (eqn\;1)

When s = 25 and t = 6:


25=16(6)+c(6)+c1=16(6)+6c+c125=16(2.45)+6c+c16c+c1=25+39.2    6c+c1=64.2(eqn  2)25 = -16(\sqrt{6})+c(6) + c_1\\ = -16(\sqrt{6}) + 6c + c_1\\ 25= -16(2.45) +6c +c_1\\ 6c+c_1 = 25+39.2 \\ \implies 6c+c_1 = 64.2 \qquad\dots (eqn\;2)

Solving eqn 1 and eqn 2 simultaneously:


4c+c1=48()6c+c1=64.22c=16.2c=8.1\quad4c+c_1=48\\ (-) 6c+c1=64.2\\ ---------\\ -2c=-16.2\\ c= 8.1

Hence we find c1c_1 :


4(8.1)+c1=4832.4+c1=48c1=4832.4c1=15.64(8.1) +c_1 = 48\\ 32.4 +c_1 = 48\\ c_1 = 48 -32.4\\ c_1=15.6

Therefore, the equation of the motion s=f(t)s = f(t) is:


s=f(t)=16t+8.1t+15.6s=f(t)=-16\sqrt{t}+8.1t+15.6

The velocity v(t)v(t) is:


v(t)=8t+8.1v(t) = -\dfrac{8}{\sqrt{t}} + 8.1


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Comments

Assignment Expert
16.03.21, 18:13

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Meep Melphy
16.03.21, 11:00

Thank you.

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