Question

 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). 

Answer 1

$v(t)=\int a(t)dt= \int 4t^{-3/2}dt= \frac{-8}{\sqrt{t}} +C_1$

$s(t)= \int v(t)dt =\int( -8t^{-0.5}+C_1)dt =-16\sqrt{t} +C_1t + C_2$

16=-32+4C₁+C₂

C₂=48+4C₁

$25=-16 \sqrt{6}+ 6C1+48+4C1$

$10C_1=-23+16 \sqrt{6}$

$C_1=1.6 \sqrt{6}-2.3$

$C_2=38.8+6.4 \sqrt{6}$

Answer: $v(t)= \frac{-8}{\sqrt{t}}+1.6\sqrt{6}-2.3$

$s(t)=-16\sqrt{t}+t(1.6 \sqrt{6}-2.3) +38.8+6.4\sqrt{6}$