Mathematics: A particle moves along a line with acceleration a (t) = -1/(t+2)2 ft/sec2. Find the distance traveled by the particle during the time interval [0, 5], given that the initial velocity v(0) is 1/2 ft/sec. -4 1n(2)+4 1n(7)ft -ln(2)+ln(7)ft -31n(2) + 31n(7) ft -21n(2) + 21n(7) ft -1/2 ln(2) + 1/2ln(7)ft

Y=3/2X+3y=mx+b3/2 = slope (m)y-y1=m(X-X1)(-2 = X1)(0 = Y1)y-0=3/2(x-(-2)y=3/2x+3...

A particle moves along a line with acceleration

kinglightskin2k

18.01.2022

$s(t)=(ln|7|+ln|2|)\,ft$

Step-by-step explanation:

Acceleration is second derivative of distance and are related as:

$a(t)=\frac{d^2s}{dt^2}\\\\\frac{d^2s}{dt^2}=\frac{-1}{(t+2)^2}\\$

Integrating both sides w.r.to t

$v(t)=\frac{ds}{dt}=\frac{1}{t+2} +C\\$

Using initial value

$v(0)=\frac{1}{2}\\\\\frac{1}{2}=\frac{1}{0+2} +C\\\\C=0\\\\\frac{ds}{dt}=\frac{1}{t+2}$

We have to calculate the distance covered in time interval [0,5], so:

$\int\limits^5_0 \frac{ds}{dt}=\int\limits^5_0 {\frac{1}{t+2}} \, dt\\\\s(t)=[ln|t+2|]^5_0\\\\s(t)=ln|5+2|+ln|0+2|\\\\s(t)=(ln|7|+ln|2|)\,ft$

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