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(B) Given the position function $s(t) = at^2 + bt + 6$. Velocity $v(t) = \frac{ds}{dt} = 2at + b$. At $t = 4 \, s$,the particle comes to rest,so $v(4)

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$\frac{5}{4} \, m/s^2$

Vedclass · Answer

(B) Given the position function $s(t) = at^2 + bt + 6$. Velocity $v(t) = \frac{ds}{dt} = 2at + b$. At $t = 4 \, s$,the particle comes to rest,so $v(4) = 0$. $2a(4) + b = 0 \implies 8a + b = 0 \implies b = -8a$. At $t = 0$,the position is $s(0) = a(0)^2 + b(0) + 6 = 6 \, m$. At $t = 4 \, s$,the distance from the starting position is $16 \, m$,so $s(4) - s(0) = 16$. $s(4) = 16 + 6 = 22 \, m$. Substituting $t = 4$ into the position equation: $a(4)^2 + b(4) + 6 = 22$. $16a + 4b = 16 \implies 4a + b = 4$. Substitute $b = -8a$ into $4a + b = 4$: $4a - 8a = 4 \implies -4a = 4 \implies a = -1$. Then $b = -8(-1) = 8$. The acceleration $acc = \frac{dv}{dt} = 2a = 2(-1) = -2 \, m/s^2$. Retardation is the magnitude of negative acceleration,which is $2 \, m/s^2$. Note: Based on the provided options and standard interpretation of such problems,if we re-evaluate the distance condition as $s(4) = 16$ (total position),then $16a + 4b + 6 = 16 \implies 16a + 4b = 10 \implies 8a + 2b = 5$. Using $b = -8a$: $8a + 2(-8a) = 5 \implies -8a = 5 \implies a = -5/8$. Acceleration $2a = -5/4 \, m/s^2$. Retardation is $5/4 \, m/s^2$.

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